arXiv:0812.1612v1 [math.QA] 9 Dec 2008 · concerning quantized coordinate rings and their...

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arXiv:0812.1612v1 [math.QA] 9 Dec 2008 SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS K. R. Goodearl Dedicated to S. K. Jain on the occasion of his 70th birthday Abstract. This paper offers an expository account of some ideas, methods, and conjectures concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of an affine algebraic variety V consists of the classical coordinate ring O(V ) equipped with an associated Poisson structure. Conjectured relationships between primitive ideals of a generic quantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit Poisson structure on O(V )) are discussed, as are breakdowns in the connections when the symplectic leaves are not algebraic. This prompts replacement of the differential-geometric concept of symplectic leaves with the algebraic concept of symplectic cores, and a reformulated conjecture is proposed: The primitive spectrum of A should be homeomorphic to the space of symplectic cores in V , and to the Poisson-primitive spectrum of O(V ). Various examples, including both quantized coordinate rings and enveloping algebras of solvable Lie algebras, are analyzed to support the choice of symplectic cores to replace symplectic leaves. 0. Introduction By now, the “Cheshire cat” description of quantum groups is well known – a quantum group is not a group at all, but something that remains when a group has faded away, leav- ing an algebra of functions behind. The appropriate functions depend on which category of group is under investigation. We concentrate here on (affine) algebraic groups G, on which the natural functions of interest are the polynomial functions. These constitute the classical coordinate ring of G, which we denote O(G). (The group structure on G induces a Hopf algebra structure on O(G), but we shall not make use of that.) A quantized coordi- nate ring of G is, informally, a deformation of O(G), in the sense that it is an algebra with a set of generators patterned after those in O(G), but with a new multiplication that is typically noncommutative. Examples and references will be given in Section 1. We do not address the question of what properties are required to qualify an algebra as a quantized coordinate ring – this remains a fundamental open problem. Quantized coordinate rings have also been defined for a number of algebraic varieties other than algebraic groups, and our discussion will incorporate them as well. 2000 Mathematics Subject Classification. 16W35; 16D60, 17B63, 20G42. Key words and phrases. Quantized coordinate ring, semiclassical limit, Poisson algebra, symplectic leaf, symplectic core, Dixmier map. Typeset by A M S-T E X 1

Transcript of arXiv:0812.1612v1 [math.QA] 9 Dec 2008 · concerning quantized coordinate rings and their...

Page 1: arXiv:0812.1612v1 [math.QA] 9 Dec 2008 · concerning quantized coordinate rings and their semiclassical limits, with a particular focus on primitive ideal spaces. The semiclassical

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS

K. R. Goodearl

Dedicated to S. K. Jain on the occasion of his 70th birthday

Abstract. This paper offers an expository account of some ideas, methods, and conjectures

concerning quantized coordinate rings and their semiclassical limits, with a particular focuson primitive ideal spaces. The semiclassical limit of a family of quantized coordinate rings of

an affine algebraic variety V consists of the classical coordinate ring O(V ) equipped with an

associated Poisson structure. Conjectured relationships between primitive ideals of a genericquantized coordinate ring A and symplectic leaves in V (relative to a semiclassical limit

Poisson structure on O(V )) are discussed, as are breakdowns in the connections when the

symplectic leaves are not algebraic. This prompts replacement of the differential-geometricconcept of symplectic leaves with the algebraic concept of symplectic cores, and a reformulated

conjecture is proposed: The primitive spectrum of A should be homeomorphic to the spaceof symplectic cores in V , and to the Poisson-primitive spectrum of O(V ). Various examples,

including both quantized coordinate rings and enveloping algebras of solvable Lie algebras,

are analyzed to support the choice of symplectic cores to replace symplectic leaves.

0. Introduction

By now, the “Cheshire cat” description of quantum groups is well known – a quantumgroup is not a group at all, but something that remains when a group has faded away, leav-ing an algebra of functions behind. The appropriate functions depend on which categoryof group is under investigation. We concentrate here on (affine) algebraic groups G, onwhich the natural functions of interest are the polynomial functions. These constitute theclassical coordinate ring of G, which we denote O(G). (The group structure on G inducesa Hopf algebra structure on O(G), but we shall not make use of that.) A quantized coordi-

nate ring of G is, informally, a deformation of O(G), in the sense that it is an algebra witha set of generators patterned after those in O(G), but with a new multiplication that istypically noncommutative. Examples and references will be given in Section 1. We do notaddress the question of what properties are required to qualify an algebra as a quantizedcoordinate ring – this remains a fundamental open problem. Quantized coordinate ringshave also been defined for a number of algebraic varieties other than algebraic groups, andour discussion will incorporate them as well.

2000 Mathematics Subject Classification. 16W35; 16D60, 17B63, 20G42.

Key words and phrases. Quantized coordinate ring, semiclassical limit, Poisson algebra, symplecticleaf, symplectic core, Dixmier map.

Typeset by AMS-TEX

1

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2 K. R. GOODEARL

Many parallels have been found between the structures of quantized and classical coor-dinate rings, and general principles for organizing and predicting such parallels are needed.The present paper concentrates on a circle of ideas and results focussed on ideal structure,particularly spaces of prime or primitive ideals. The theme/principle we follow, based onmuch previous work, can be stated this way:

• The primitive ideals of a suitably generic quantized coordinate ring of an algebraicvariety V should match subsets of V in some partition defined through the geometryof V and a Poisson structure obtained from a semiclassical limit process.

Many of the terms just mentioned require explanations, which we will give over the courseof the paper. Here we just mention that, in the above statement, “generic” refers to theassumption that suitable parameters in the construction of the quantized coordinate ringshould be non-roots of unity.

To begin the story (omitting many definitions and details), we refer to the resultsof Soibelman and Vaksman [51, 45, 46], who studied the “standard” generic quantizedcoordinate rings of simple compact Lie groups K. They established a bijection betweenthe irreducible *-representations of K (on Hilbert spaces) and the symplectic leaves in K(relative to a Poisson structure arising from the quantization). This amounts to a linkagebetween primitive ideals and symplectic leaves, a relationship which is a key ingredient ofthe Orbit Method from Lie theory. Informed by this principle, and inspired by the work ofSoibelman and Vaksman, Hodges and Levasseur conjectured that similar bijections shouldexist for semisimple complex algebraic groups [22]. The case of SL2(C) being easy [22,Appendix], they first verified the conjecture for SL3(C) [op. cit.], and then for SLn(C)[23]. In later work with Toro [24], they verified it for connected semisimple groups. Inlight of these achievements, it is natural to pose this conjecture for other classes of genericquantized coordinate rings. (It is easily seen that the above conjecture cannot hold fornon-generic quantized coordinate rings. In such cases, the quantized coordinate rings areusually finitely generated modules over their centers, and they have far more primitiveideals than can be matched to symplectic leaves.)

In the specific cases just mentioned, the symplectic leaves turn out to be algebraic, inthe sense that they are locally closed in the Zariski topology. Hodges, Levasseur, and Toropointed out in [24] that symplectic leaves need not be algebraic for Poisson structuresarising from multiparameter quantizations, and that the above conjecture cannot be ex-pected to hold in such cases. We argue that it should not be surprising that the conceptof symplectic leaves, which comes from differential geometry, is not always well suited foralgebraic problems. Thus, symplectic leaves should be replaced by more algebraically de-fined objects. The notion of symplectic cores introduced by Brown and Gordon [6] fillsthe role well, up to the present state of knowledge; we will give evidence to buttress thisstatement.

Our aim here is to present an account of the above story, with introductions to anddiscussions of the relevant concepts. In particular, the tour will pass through way stationssuch as quantized coordinate rings , semiclassical limits , Poisson structures , symplectic

leaves , the Orbit Method , symplectic cores , and the Dixmier map. By the end of the tour,

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 3

we will be in purely algebraic territory, where we can formulate a conjecture that does notrequire any differential geometry (i.e., symplectic leaves). Namely:

• If A is a generic quantized coordinate ring of an affine algebraic variety V over analgebraically closed field of characteristic zero, and if V is given the Poisson struc-ture arising from an appropriate semiclassical limit, then the spaces of primitiveideals in A and symplectic cores in V , with their respective Zariski topologies, arehomeomorphic.

A parallel conjecture relates the prime and primitive spectra of A to the spaces of Poissonprime and Poisson-primitive ideals in O(V ).

Fix a base field k throughout the paper; all algebras mentioned will be unital k-algebras.This field can be general at first, but then we will require it to have characteristic zero,and/or be algebraically closed. When discussing symplectic leaves, we restrict k to R orC.

1. Quantized coordinate rings

We begin by recalling two basic examples, to clarify the idea that a quantized coordinatering of an algebraic group (or variety) is, loosely speaking, a deformation of the classicalcoordinate ring. References to many other examples are given in §§1.2, 1.4, 1.5.1.1. Quantum SL2. Recall that the group SL2(k) is a closed subvariety of the variety of2× 2 matrices over k, defined by the single equation “determinant = 1”. The coordinatering of the matrix variety is naturally realized as a polynomial ring in four variablesXij , corresponding to the functions that pick out the four entries of the matrices. Thecoordinate ring of SL2(k) can thus be described as follows:

O(SL2(k)) = k[X11, X12, X21, X22]/〈X11X22 −X12X21 − 1〉.

To “quantize” this coordinate ring, we replace the commutative multiplication by a non-commutative one, parametrized by a nonzero scalar q, as below. The reasons for thisparticular choice of relations will not be given here; see [4, §§I.1.6, I.1.8], for instance, fora discussion.

Given a choice of scalar q ∈ k×, the “standard” one-parameter quantized coordinate

ring of SL2(k) is the k-algebra Oq(SL2(k)) presented by generators X11, X12, X21, X22

and the following relations:

X11X12 = qX12X11 X11X21 = qX21X11

X12X22 = qX22X12 X21X22 = qX22X21

X12X21 = X21X12 X11X22 −X22X11 = (q − q−1)X12X21

X11X22 − qX12X21 = 1 .

The case when q = 1 is special: The first six relations then reduce to saying that thegenerators Xij commute with each other, the last reduces to the defining relation for the

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4 K. R. GOODEARL

variety SL2(k), and so the algebra O1(SL2(k)) is just the classical coordinate ring. Wewrite this, very informally, as

O(SL2(k)) = limq→1Oq(SL2(k));

it is our first example of a “semiclassical limit”.

1.2. Quantum matrices, quantum SLn and GLn. The pattern indicated in §1.1 ex-tends to definitions of “standard” single parameter quantized coordinate rings Oq(Mn(k)),Oq(SLn(k)), and Oq(GLn(k)) for all positive integers n. Multiparameter versions, whichwe label in the form Oλ,p(−), have also been defined. Generators and relations for thesealgebras may be found, for instance, in [12, §§1.2–1.4; 4, §§I.2.2–I.2.4].

1.3. Quantum affine spaces. The coordinate ring of affine n-space over k is the polyno-mial algebra in n indeterminates, and the most basic quantization is obtained by replacingcommutativity (xy = yx) with q-commutativity : xy = qyx. Thus, the “standard” one-parameter quantized coordinate ring of kn, relative to a choice of scalar q ∈ k×, is thek-algebra

Oq(kn) = k〈x1, . . . , xn | xixj = qxjxi for 1 ≤ i < j ≤ n〉.

The multiparameter version of this algebra requires an n × n matrix of nonzero scalars,q = (qij), which is multiplicatively antisymmetric in the sense that qii = 1 and qji = q−1

ij

for all i, j. The multiparameter quantized coordinate ring of kn corresponding to a choiceof q is the k-algebra

Oq(kn) = k〈x1, . . . , xn | xixj = qijxjxi for all i, j〉.

In the one-parameter case, we can write O(kn) = limq→1

Oq(kn) in the same sense as

above. For the multiparameter case, we imagine a limit in which all qij → 1.

1.4. Quantized coordinate rings of semisimple groups. The single parameter ver-sions of these Hopf algebras, which we denote Oq(G), were first defined for semisimplealgebraic groups G of classical type (types A, B, C, D) via generators and relations, byFaddeev, Reshetikhin, and Takhtadjan [43] and Takeuchi [48]. A detailed development(done for k = C, but the pattern is the same over other fields) can be found in [32, Chapter9]. In most of the more recent literature, Oq(G) is defined as a restricted Hopf dual ofthe quantized enveloping algebra of the Lie algebra of G (e.g., see [4, Chapter I.7]). Thisis a more uniform approach, which also covers groups of exceptional type. That the twoapproaches yield the same Hopf algebras in the classical cases was established by Hayashi[20] and Takeuchi [48] (see [32, Theorem 11.22]).

The single parameter algebras Oq(G) constitute the “standard” quantized coordinaterings of semisimple groups. Multiparameter versions, which we label Oq,p(G), were intro-duced by Hodges, Levasseur, and Toro [24].

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 5

1.5. Additional examples. Quantized coordinate rings, both single- and multiparame-ter, have been defined for many algebraic varieties, such as algebraic tori, toric varieties,and versions of affine spaces related to classical groups of types B, C, D. For a generalsurvey, see [12, Section 1]. Quantized toric varieties were introduced in [26] (see also [17,16]). A family of iterated skew polynomial algebras covering multiparameter quantizedeuclidean and symplectic spaces was introduced by Oh [38] and extended by Horton [25](see also [14, §2.5] for the odd-dimensional euclidean case). Among other algebras thathave been studied in the literature, we mention quantized coordinate rings for varieties ofantisymmetric matrices [47] and varieties of symmetric matrices [37, 28].

1.6. Limits of families of algebras. The semiclassical limits informally introduced in§§1.1, 1.3 are more properly viewed in the framework of families of algebras. For example,the algebras Oq(SL2(k)) are quotients of a single algebra over a Laurent polynomial ringk[t±1], namely the algebra A given by generators X11, X12, X21, X22 and relations as in§1.1, but with q replaced by t:

(1.6a)

X11X12 = tX12X11 X11X21 = tX21X11

X12X22 = tX22X12 X21X22 = tX22X21

X12X21 = X21X12 X11X22 −X22X11 = (t− t−1)X12X21

X11X22 − tX12X21 = 1 .

For each q ∈ k×, there is a natural identification A/(t− q)A ≡ Oq(SL2(k)). The “limit asq → 1” is then simply the case q = 1 of these identifications: A/(t− 1)A ≡ O(SL2(k)).

Similarly, if we take

(1.6b) B = k[t±1]〈x1, . . . , xn | xixj = txjxi for 1 ≤ i < j ≤ n〉,

then B/(t− q)B ≡ Oq(kn) for all q ∈ k×, and

limq→1Oq(k

n) = B/(t− 1)B ≡ O(kn).

The multiparameter algebras Oq(kn) can, likewise, be set up as common quotients of an

algebra over a Laurent polynomial ring k[t−1ij | 1 ≤ i < j ≤ n]. However, for purposes such

as obtaining Poisson structures on semiclassical limits, we need to be able to exhibit theOq(k

n) as quotients of k[t±1]-algebras. There are many ways to do this; we will discusssome in §2.3.1.7. An older example: the Weyl algebra. Weyl defined the algebra we now call thefirst Weyl algebra as

C〈x, y | xy − yx = ℏi〉,where ℏ is Planck’s constant and i =

√−1. Physicists often use the term “classical limit”

to denote the transition from a quantum mechanical system to a classical one by lettingPlanck’s constant go to zero. The fact that lim

ℏ→0of the above algebra is the polynomial

ring C[x, y] is one instance of this point of view.

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6 K. R. GOODEARL

To relate this semiclassical limit to the ones above, take k = C and take the scalarq in quantized coordinate rings to be eℏ. Then ℏ → 0 corresponds to q → 1. In manyconstructions, particularly the C*-algebra quantum groups corresponding to compact Liegroups, the parameter q is either written directly in the form eℏ or is taken to be anonnegative real number, with calculations involving eℏ used for motivation.

2. Semiclassical limit constructions

In the context of quantized coordinate rings, semiclassical limits are constructed viaquotients of algebras over Laurent polynomial rings, as in §1.6. A different version, usingassociated graded rings, is needed in other arenas, particularly for enveloping algebras ofLie algebras. We describe both constructions in this section.

2.1. Semiclassical limits: commutative fibre version. Let k[h] be a polynomialalgebra, with the indeterminate named h as a reminder of Planck’s constant. Supposethat A is a torsionfree k[h]-algebra, and that A/hA is commutative. Since A is then a flatk[h]-module, the family of factor algebras

(A/(h−α)A

)α∈k

(or, for short, A itself) is called

a flat family of k-algebras, and A/hA is viewed as the “limit” of the family. It may happenthat some of the algebras A/(h − α)A collapse to zero or are otherwise not desirable. Ifso, it is natural to treat A as an algebra over a localization of k[h] (cf. Example 2.2(c), forinstance). We will usually not do this explicitly.

An immediate question is, what kind of information about the algebras A/(h− α)A iscontained in this limit? Observe that, because of the commutativity of A/hA, all additivecommutators [a, b] = ab − ba in A are divisible by h. Moreover, division by h is unique,since A is torsionfree as a k[h]-module. Hence, we obtain a well defined binary operation1h[−,−] on A. This operation enjoys four key properties:

(1) Bilinearity;(2) Antisymmetry;(3) The Jacobi identity (thus A, equipped with 1

h[−,−], is a Lie algebra over k);

(4) The Leibniz identities , that is, the product rule (for derivatives) in each variable:1h[a, bc] =

(1h[a, b]

)c+ b

(1h[a, c]

)for all a, b, c ∈ A, and similarly for 1

h[bc, a].

Operations satisfying properties (1)–(4) are called Poisson brackets .The above Poisson bracket on A induces, uniquely, a Poisson bracket on A/hA, which

we denote {−,−}. Thus, writing overbars to denote cosets modulo hA, we have

{a, b} = 1h[a, b]

for a, b ∈ A. The commutative algebra A/hA, equipped with this Poisson bracket, is calledthe semiclassical limit of the family

(A/(h − α)A

)α∈k

. Loosely speaking, the Poisson

bracket on the semiclassical limit records a “first-order impression” of the commutators inA and in the algebras A/(h− α)A.2.2. Examples. (a) Fit the one-parameter quantum affine spaces Oq(k

n) into the k[t±1]-algebra B of (1.6b), and set h = t − 1. Then B represents a flat family of k[h]-algebras,with B/hB commutative. We identify B/hB with the polynomial ring k[x1, . . . , xn] and

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 7

compute the resulting Poisson bracket on the indeterminates as follows. For 1 ≤ i < j ≤ n,we have [xi, xj] = hxjxi in B, and hence

{xi, xj} = xixj

in k[x1, . . . , xn]. Because of the Leibniz identities, the above information determines thisPoisson bracket uniquely. It may be described in full as follows:

{f, g} =∑

1≤i<j≤n

xixj

(∂f

∂xi

∂g

∂xj− ∂g

∂xi

∂f

∂xj

)

for all f, g ∈ k[x1, . . . , xn].(b) Take A = k[h]〈x, y | xy − yx = h〉. Then A/(h − α)A ∼= A1(k) for all nonzero

α ∈ k, while A/hA can be identified with the polynomial ring k[x, y]. In this case, thesemiclassical limit Poisson bracket on k[x, y] satisfies (and is determined by)

{x, y} = 1.

(c) The family(Oq(SL2(k))

)q∈k×

fits into the k[t±1]-algebra A with generators X11,

X12, X21, X22 and relations (1.6a). This is a flat family over k[h], where h = t− 1. Sincet is invertible in A, the specialization A/(h+1)A is zero, corresponding to the fact that Ais actually a torsionfree (even free) algebra over the localization k[h][(h+ 1)−1]. Here thesemiclassical limit is the classical coordinate ring O(SL2(k)), equipped with the Poissonbracket satisfying

{X11, X12} = X11X12 {X11, X21} = X11X21

{X12, X22} = X12X22 {X21, X22} = X21X22

{X12, X21} = 0 {X11, X22} = 2X12X21 .

2.3. Multiparameter examples. To obtain a semiclassical limit – with Poisson bracket– for a multiparameter family of algebras, we convert to a single parameter family andapply the construction of §2.1. The procedure is clear when the parameters involved areinteger powers of a single parameter. For example, consider the algebras Oq(k

n) whereq = (qaij ) for q ∈ k× and an antisymmetric integer matrix (aij). Then define

A = k[t±1]〈x1, . . . , xn | xixj = taijxjxi for all i, j〉,

which is a torsionfree k[t − 1]-algebra with A/(t − 1)A commutative. The semiclassicallimit is the polynomial algebra k[x1, . . . , xn], equipped with the Poisson bracket satisfying

{xi, xj} = aijxixj

for all i, j.

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8 K. R. GOODEARL

More general parameters can be dealt with by various means. A simple but ad hocmethod to handle any Oq(k

n) is via the algebra

A = k[h]〈x1, . . . , xn | xixj =(1 + (qij − 1)h

)xjxi for 1 ≤ i < j ≤ n〉,

which is set up so that A/(h − 1)A ∼= Oq(kn) and A/hA ∼= k[x1, . . . , xn]. This yields a

Poisson bracket satisfying {xi, xj} = (qij − 1)xixj for all i, j.A variant of the previous procedure, involving quadratic rather than linear polynomials

in h, is used in [18] to construct semiclassical limits for which the conjecture sketched inthe Introduction applies to the generic multiparameter quantum affine spaces Oq(k

n).Inverse to the construction of semiclassical limits is the problem of quantization: trying

to represent a given algebra supporting a Poisson bracket as a semiclassical limit of asuitable family of algebras. We will not discuss this problem except to indicate a solutionfor the case of homogeneous quadratic Poisson brackets on polynomial rings. Namely,suppose we have a polynomial algebra k[x1, . . . , xn], equipped with a Poisson bracket suchthat {xi, xj} = αijxixj for all i, j, where (αij) is an antisymmetric matrix of scalars overk. In place of ad hoc procedures such as the one sketched above, it is natural, assumingthat char k = 0, to use power series. In this case, set exp(αij) =

∑∞

n=01n!α

nijh

n ∈ k[[h]] forall i, j, and form the k[[h]]-algebra

A = k[[h]]〈x1, . . . , xn | xixj = exp(αij)xjxi for all i, j〉.

The semiclassical limit algebra is k[x1, . . . , xn], and its Poisson bracket is the original one.Analogous k[[h]]-algebra constructions are given for commonly studied families of quan-

tized coordinate rings of skew polynomial type in [14, Section 2].

2.4. Semiclassical limits: filtered/graded version. Suppose that A is a Z-filteredk-algebra, say with filtration (An)n∈Z. Thus, the An are k-subspaces of A, with Am ⊆ An

when m ≤ n, such that AmAn ⊆ Am+n for all m, n. We will assume that the filtrationis exhaustive, that is, that

⋃n∈Z

An = A. Note that we must have 1 ∈ A0; thus, A0 is aunital subalgebra of A. Finally, let grA =

⊕n∈Z

grnA be the associated graded algebra,where grnA = An/An−1.

Now assume that grA is commutative. Homogeneous elements a ∈ grmA and b ∈grnA can be lifted to elements a ∈ Am and b ∈ An, and since grA is commutative, the

commutator [a, b] must lie in Am+n−1. We then set {a, b} equal to the coset of [a, b] ingrm+n−1A. It is an easy exercise, left to the reader, to verify that {a, b} is well defined, andthat the extension of {−,−} to sums of homogeneous elements defines a Poisson bracketon grA. The commutative algebra grA, equipped with this Poisson bracket, is called thesemiclassical limit of A.

More generally, assume there is an integer d < 0 such that [Am, An] ⊆ Am+n+d forall m,n ∈ Z. This assumption of course forces grA to be commutative. Modify the

definition above by setting {a, b} equal to the coset of [a, b] in grm+n+dA, for a ∈ grmAand b ∈ grnA. This recipe again produces a well defined Poisson bracket on grA [34,Lemma 2.7].

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 9

2.5. Bridging the two constructions. The semiclassical limit of a Z-filtered algebraA constructed in §2.4 can also be obtained by applying the construction of §2.1 to anauxiliary algebra, namely the Rees ring

A :=∑

n∈Z

Anhn ⊆ A[h±1],

where A[h±1] is a Laurent polynomial ring over A. Since 1 ∈ A0, the polynomial algebra

k[h] is a subalgebra of A, and we note that A is a torsionfree k[h]-algebra. (It is not a k[h±1]-

algebra unless A−1 = A0, in which case all An = A0.) On one hand, A/(h − 1)A ∼= A.

On the other, A/hA ∼= grA, because hA =∑

n∈ZAn−1h

n. Thus, if grA is commutative,

we have a Poisson bracket 1h[−,−] on A, which induces a Poisson bracket {−,−}1 on grA

as in §2.1. This bracket concides with the Poisson bracket {−,−}4 constructed in §2.4, asfollows.

Start with a ∈ grmA and b ∈ grnA, and lift these elements to a ∈ Am and b ∈ An.

With respect to the natural epimorphism π : A → grA, the elements a and b lift to

ahm, bhn ∈ A. Hence,

{a, b}1 = π(1h[ahm, bhn]

)= π([a, b]hm+n−1) = [a, b] +Am+n−2 = {a, b}4 .

Therefore {−,−}1 = {−,−}4.2.6. Example: enveloping algebras. Let g be a finite dimensional Lie algebra overk, and put the standard (nonnegative) filtration on the enveloping algebra U(g), so thatU(g)0 = k and U(g)1 = k + g, while U(g)n = U(g)n1 for n > 1. The associated gradedalgebra is commutative, and is naturally identified with the symmetric algebra S(g) of thevector space g. In particular, we use the same symbol to denote an element of g and itscoset in gr1 U(g) = S(g)1. Then S(g) is the semiclassical limit of U(g), equipped with thePoisson bracket satisfying

{e, f} = [e, f ]

for all e, f ∈ g, where [e, f ] denotes the Lie product in g. The above formula determines{−,−} uniquely, since g generates S(g).

Now view the dual space g∗ as an algebraic variety, namely the affine space Adim g. Thecoordinate ring O(g∗) is a polynomial algebra over k in dim g indeterminates, as is S(g).There is a canonical isomorphism

(2.6) θ : S(g)∼=−−→ O(g∗)

which sends each e ∈ g to the polynomial function on g∗ given by evaluation at e, thatis, θ(e)(α) = α(e) for α ∈ g∗. (This isomorphism is often treated as an identificationof the algebras S(g) and O(g∗).) Via θ, the Poisson bracket on S(g) obtained from thesemiclassical limit process above carries over to a Poisson bracket on O(g∗), known as theKirillov-Kostant-Souriau Poisson bracket .

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10 K. R. GOODEARL

If {e1, . . . , en} is a basis for g, then S(g) = k[e1, . . . , en] and θ sends the ei to indeter-minates xi such that O(g∗) = k[x1, . . . , xn]. An explicit description of the KKS Poissonbracket on O(g∗) can be obtained in terms of the structure constants of g, as follows. Theseconstants are scalars clij ∈ k such that [ei, ej] =

∑l c

lijel for all i, j. Since {ei, ej} = [ei, ej]

in S(g), an application of θ yields {xi, xj} =∑

l clijxl for all i, j. It follows that

{p, q} =∑

i,j,l

clijxl∂p

∂xi

∂q

∂xj

for p, q ∈ O(g∗) [7, Proposition 1.3.18]. To see this, just check that the displayed formuladetermines a Poisson bracket on O(g∗) which agrees with the KKS bracket on pairs ofindeterminates.

The KKS Poisson bracket on O(g∗) can also be obtained by applying the method of §2.1to the homogenization of U(g), that is, the k[h]-algebra A with generating vector space g

and relations ef − fe = h[e, f ] for e, f ∈ g (where [e, f ] again denotes the Lie product ing). Here A/hA ∼= S(g) ∼= O(g∗) and A/(h− λ)A ∼= U(g) for all λ ∈ k×.

3. Symplectic leaves

We introduce symplectic leaves first in the context of Poisson manifolds, following theoriginal definition of Weinstein [54], and then we carry the concept over to complex affinePoisson varieties, following Brown and Gordon [6].

3.1. Poisson algebras. We reiterate the general definition from §2.1: a Poisson bracket

on a k-algebra R is any antisymmetric bilinear map R×R→ R which satisfies the Jacobiand Leibniz identities. Unless a special notation imposes itself, we denote all Poissonbrackets by curly braces: {−,−}.

A Poisson algebra over k is just a k-algebra R equipped with a particular Poissonbracket. We restrict our attention to commutative Poisson algebras in the present paper.As for the noncommutative case, Farkas and Letzter have shown that Poisson bracketsessentially reduce to commutators [11, Theorem 1.2]: If R is a prime ring which is notcommutative, any Poisson bracket on R is a multiple of the commutator bracket by anelement of the extended centroid of R.

3.2. Symplectic leaves in Poisson manifolds. Let M be a smooth manifold, andlet C∞(M) denote the algebra of smooth real-valued functions on M . (Some authorsreplace C∞(M) by the algebra of smooth or analytic complex-valued functions.) A Poisson

structure on M is a choice of Poisson bracket on C∞(M), so that C∞(M) becomes aPoisson algebra. A smooth manifold, together with a choice of Poisson structure, is calleda Poisson manifold .

Now assume thatM is a Poisson manifold. For each f ∈ C∞(M), the map Xf = {f,−}is a derivation on C∞(M) and thus a vector field on M . Such vector fields are calledHamiltonian vector fields (for the given Poisson structure), and the flows (or integralcurves) of Hamiltonian vector fields are known as Hamiltonian paths . More specifically,a smooth path γ : [0, 1] → M is Hamiltonian provided there is some f ∈ C∞(M) such

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 11

that, at each point γ(t) along the path, the tangent vector dγ/dt equals Xf |γ(t). Since thechange from a Hamiltonian path following the flow of a vector field Xf to one following adifferent vector field Xg need not be smooth, one must work with piecewise Hamiltonian

paths , i.e., finite concatenations of Hamiltonian paths.These paths determine an equivalence relation onM , points p and p′ being equivalent if

and only if there is a piecewise Hamiltonian path in M running from p to p′. The resultingequivalence classes are called symplectic leaves , and the partition of M as the disjointunion of its symplectic leaves is known as the symplectic foliation of M .

3.3. Poisson bivector fields. For many purposes, it is more useful to record a Poissonstructure in the form of a bivector field rather than a Poisson bracket. In particular, thisallows the most direct definition of Poisson structures on non-affine algebraic varieties.

Let M be a Poisson manifold. For a point p ∈ M , let mp denote the maximal ideal ofC∞(M) consisting of those functions that vanish at p. Evaluation of Poisson brackets atp induces an antisymmetric bilinear form πp on the cotangent space mp/m

2p, where

πp(f +m2p, g +m2

p) = {f, g}(p)

for f, g ∈ mp. Now πp acts in each variable as a linear map in the dual space of mp/m2p, that

is, as a tangent vector to M at p. Since πp is antisymmetric, it is thus a tangent bivector

at p, namely an element of Tp(M) ∧ Tp(M). The map π : p 7→ πp is a smooth globalsection of Λ2TM , that is, a tangent bivector field on M . To recover the Poisson bracketon C∞(M) from the bivector field π, observe that {f, g}(p) = {f − f(p), g − g(p)}(p) forf, g ∈ C∞(M) and p ∈M , which we rewrite in the form

(3.3) {f, g}(p) = πp(df(p), dg(p)),

where df(p) = f − f(p) +m2p ∈ mp/m

2p and similarly for dg(p).

Conversely, via (3.3) any tangent bivector field π onM induces an antisymmetric bilinearmap {−,−} on C∞(M) satisfying the Leibniz conditions. This is a Poisson bracket exactlywhen the Jacobi identity is satisfied, which is equivalent to the vanishing of the Schouten

bracket [π, π] (which we will not define here; see [1, p. 44; 53, 2nd. ed., Remark 2.2(3)],for instance). A Poisson bivector field on M is any tangent bivector field π for which[π, π] = 0. As indicated in the sketch above, Poisson brackets on C∞(M) correspondbijectively to Poisson bivector fields on M .

3.4. Poisson varieties. For any complex algebraic variety V , the definition of a Poisson

bivector field on V can be copied from §3.3 – it is any tangent bivector field π on V forwhich [π, π] = 0. In the context of algebraic geometry, however, the map π : V → Λ2TVis required to be a regular function. Now one defines a Poisson variety to be a complexalgebraic variety equipped with a particular Poisson bivector field. Associated conceptsare defined by requiring compatibility with these bivector fields. For example, a Poisson

morphism from a Poisson variety (V, π) to a Poisson variety (W,π′) is a regular mapφ : V → W such that (Tφ ∧ Tφ)π = π′φ. A Poisson subvariety of V is a subvariety Xsuch that the inclusion map X → V is a Poisson morphism.

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12 K. R. GOODEARL

If V is an affine Poisson variety, the formula (3.3) defines a Poisson bracket on O(V ).Conversely, any Poisson bracket on O(V ) induces a Poisson bivector field on V as in §3.3.Thus, affine Poisson varieties can equally well be defined as complex affine varieties whosecoordinate rings are Poisson algebras. This point of view can be extended to arbitraryvarieties by defining a Poisson variety to be a complex algebraic variety whose sheaf ofregular functions is a sheaf of Poisson algebras.

3.5. Smooth Poisson varieties as manifolds. In order to define symplectic leaves inPoisson varieties, manifold structures are needed. The fundamental result is that anysmooth (i.e., nonsingular) complex variety V has a unique structure as a complex analyticmanifold (e.g., [44, Chapter II, §2.3]). This allows one to view V as a smooth manifold.If V is a Poisson variety, its chosen Poisson bivector field π is necessarily smooth (becauseit is regular), and so V together with π becomes a Poisson manifold. One can achieve thisresult with Poisson brackets as well, by showing that any Poisson bracket on O(V ) extendsuniquely to a Poisson bracket on the algebra of smooth complex functions on V ; takingreal parts then yields a Poisson bracket on C∞(V ).

Given a smooth Poisson variety V , we view V as a smooth manifold as above, anddefine the symplectic leaves of V to be the symplectic leaves of the manifold V , defined asin §3.2.3.6. Symplectic leaves in singular Poisson varieties. Let V be an arbitrary complexvariety, and define the sequence of closed subvarieties

V0 = V ⊃ V1 ⊃ · · · ⊃ Vm = ∅,

where each Vi+1 is the singular locus of Vi. To build this chain, recall first that thesingular locus of a nonempty variety is a proper closed subvariety. Since V is a noetheriantopological space, the chain must eventually reach the empty set.

If V is a Poisson variety, then V1 is a Poisson subvariety [42, Corollary 2.4]. By induc-tion, all the Vi are Poisson subvarieties of V . Consequently, V is (canonically) the disjointunion of smooth locally closed Poisson subvarieties Zi := Vi−1 \ Vi. Following [6, §3.5], wedefine the symplectic leaves of V to be the symplectic leaves of the various Zi, defined asin §3.5.3.7. Example. There is a known recipe, described in [22, Appendix A], for determiningthe symplectic leaves in a semisimple complex algebraic group G, relative to the Poissonstructure arising from the “standard quantization” of G. For illustration, we present thecase G = SL2(C); details are given in [22, Theorem B.2.1]. The Poisson bracket on O(G)is described in §2.2(c) above. The symplectic leaves in G are as follows:

• the singletons

{[α 00 α−1

]}, for α ∈ C×;

• the sets

{[α 0γ α−1

] ∣∣∣∣ α, γ ∈ C×

}and

{[α β0 α−1

] ∣∣∣∣ α, β ∈ C×

};

• the sets

{[α βγ δ

]∈ G

∣∣∣∣ β = λγ 6= 0

}, for λ ∈ C×.

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 13

3.8. Example. The standard example of a non-algebraic solvable Lie algebra is a 3-dim-ensional complex Lie algebra g with basis {e1, e2, e3} such that

[e1, e2] = e2 [e1, e3] = αe3 [e2, e3] = 0

for some α ∈ R \Q. Write O(g∗) = C[x1, x2, x3] following the notation of §2.6. The KKSPoisson structure on g∗ is given by the Poisson bracket on O(g∗) such that

{x1, x2} = x2 {x1, x3} = αx3 {x2, x3} = 0.

As in [53, 1st. ed., Example II.2.37; 2nd. ed., Example II.2.43], the symplectic leaves in g∗

are the following sets:

• the individual points on the x1-axis;• the x1x2-plane minus the x1-axis;• the x1x3-plane minus the x1-axis;• the surfaces (x3 = λxα2 6= 0) for λ ∈ C×.

Since α is irrational, the surfaces (x3 = λxα2 6= 0) are not algebraic – they are locally closedin the euclidean topology but not in the Zariski topology.

4. The Orbit Method from Lie theory

4.1. The Orbit Method. This term has been applied to a whole complex of methodsin the representation theory of Lie groups and Lie algebras, and extended, as a guidingprinciple, to many other domains. To quote Kirillov’s survey article [30],

The idea behind the orbit method is the unification of harmonic analysis withsymplectic geometry (and it can also be considered as a part of the more generalidea of the unification of mathematics and physics). In fact, this is a post factumformulation. Historically, the orbit method was proposed in [29] for the descrip-tion of the unitary dual (i.e. the set of equivalence classes of unitary irreduciblerepresentations) of nilpotent Lie groups. It turned out that not only this problembut all other principal questions of representation theory—topological structureof the unitary dual, explicit description of the restriction and induction functors,character formulae, etc.—can be naturally answered in terms of coadjoint orbits.

In Lie theory, the relevant orbits are defined as follows. Recall that if G is a Lie groupwith Lie algebra g, then G acts on g by the adjoint action and on g∗ by the coadjoint action.The G-orbits of these actions are called adjoint orbits and coadjoint orbits , respectively. Asa particular instance, the Orbit Method suggests that the primitive ideals of the envelopingalgebra of g, being the kernels of the irreducible representations, should be related to thecoadjoint orbits in g∗. Kirillov’s original work provided the best such relationship – abijection – when g is nilpotent. There is also a bijection in case g is solvable, except thatthe coadjoint orbits may have to be taken with respect to a different group than a Liegroup with Lie algebra g. We discuss this situation in Section 5.

To place the coadjoint orbits in a geometric setting, view g∗ as the variety Adim g, as in§2.6. We can then ask for a geometric description of these orbits within g∗. The answer is

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14 K. R. GOODEARL

a famous result discovered independently by Kirillov, Kostant, and Souriau (see, e.g., [31,§I.2.2, Theorem 2]):

4.2. Theorem. [Kirillov-Kostant-Souriau] Let G be a Lie group and g its Lie algebra.

Then the coadjoint orbits of G in g∗ are precisely the symplectic leaves for the KKS Poisson

structure.

4.3. Example. Return to Example 3.8, and place the x1x2x3-coordinates of points of g∗

in column vectors. We choose a Lie group G with Lie algebra g as follows:

G :=

1 u v0 t 00 0 tα

∣∣∣∣ u, v ∈ C, t ∈ C×

.

The coadjoint action of G on g∗ can be identified with left multiplication of matrices fromG on column vectors representing points in g∗. One easily checks that the G-orbits areexactly the symplectic leaves of g∗ identified in Example 3.8, as required by Theorem 4.2.

4.4. A general principle. In situations outside Lie theory, there may not be a suitablegroup action whose orbits play the role of coadjoint orbits. Instead, taking account ofTheorem 4.2, one focusses on symplectic leaves. Restricting to the study of irreduciblerepresentations and primitive ideals, one is led to a general principle that we formulate asfollows:

Given a noncommutative algebra A, relate the primitive ideals of A to the sym-plectic leaves corresponding to the Poisson structure on some associated algebraicvariety arising from a semiclassical limit.

This loose phrasing is intended to give the flavor of ideas coming out of the Orbit Methodrather than to set up a precise recipe. Furthermore, this principle already requires modi-fication in the case of enveloping algebras, and for general quantized coordinate rings.

On the other hand, the principle is right on target for the generic single parameterquantized coordinate ringsOq(G) of semisimple complex algebraic groupsG, as conjecturedby Hodges and Levasseur in [22, §2.8, Conjecture 1]: there is a bijection between the set ofprimitive ideals of Oq(G) and the set of symplectic leaves in G (for the semiclassical limitPoisson structure). They verified this conjecture for G = SL2(C) and G = SL3(C) in [22,Corollary B.2.2, Theorems 4.4.1, A.3.2], and then for G = SLn(C) in [23, Theorem 4.2and following remarks]. For arbitrary connected, simply connected, semisimple complexLie groups G, Joseph proved in [27, Theorem 9.2] that the primitive ideal space of Oq(G)has the form conjectured by Hodges and Levasseur in the first part of [22, §2.8, Conjecture1], but he did not address connections with symplectic leaves. The full conjecture wasestablished by Hodges, Levasseur, and Toro [24, Theorems 1.8, 4.18, Corollary 4.5] forconnected semisimple complex Lie groups G. Their results also cover the multiparameteralgebra Oq,p(G) under suitable algebraicity conditions on p.

4.5. Generic versus non-generic situations. As mentioned in the introduction, theprinciple discussed in §4.4 does not apply to non-generic quantized coordinate rings, which

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 15

typically have “too many” primitive ideals. The quantum plane provides the simplestillustration of this difficulty, and of the differences between the generic and non-genericcases. Take

Aq = Oq(C2) = C〈x, y | xy = qyx〉,

where q is an arbitrary nonzero scalar in C. By Example 2.2(a), the semiclassical limitof the family (Aq)q∈C× is the polynomial ring k[x, y], equipped with the Poisson bracketsuch that {x, y} = xy. It is easily checked that the corresponding symplectic leaves in C2

consist of

• the individual points on the x- and y-axes;• the xy-plane minus the x- and y-axes.

If q is not a root of unity, one similarly checks that the primitive ideals of Aq consist of

• the maximal ideals 〈x− α, y〉 and 〈x, y − β〉, for α, β ∈ C;• the zero ideal.

(See [4, Example II.7.2], for instance, for details.) In this case, there is a natural bijectionbetween the set of primitive ideals of Aq and the set of symplectic leaves in C2.

On the other hand, if q is a primitive l-th root of unity, the center of Aq equals thepolynomial ring C[xl, yl], and Aq is a finitely generated C[xl, yl]-module. In this case,the primitive ideals of Aq are maximal ideals, and they are parametrized (up to l-to-one)by the maximal ideals of C[xl, yl]. While the set of primitive ideals of Aq has the samecardinality as the set of symplectic leaves in C2, there is no natural bijection, and certainlyno homeomorphism if Zariski topologies are taken into account.

Such disparities occur in all the standard families of quantized coordinate rings, andprovide just one of many distinctions between the generic and non-generic cases. We donot discuss the non-generic situation further, and concentrate on generic algebras.

5. Limitations of the Orbit Method for solvable Lie algebras

For a solvable finite dimensional complex Lie algebra g, the primitive ideals of theenveloping algebra U(g) are parametrized by means of the famous Dixmier map. At firstglance, this is a successful instance of the Orbit Method, since the Dixmier map induces abijection from a set of orbits in g∗ onto the set of primitive ideals of U(g). However, therelevant orbits are not, in general, those of the coadjoint action of a Lie group with Liealgebra g. Instead, the following group is needed.

5.1. The algebraic adjoint group. Let g be a finite dimensional complex Lie algebra.Treating g for a moment just as a vector space, we have the general linear group GL(g)on g, which is a complex algebraic group whose Lie algebra is the general linear Liealgebra gl(g). Any algebraic subgroup of GL(g) (i.e., any Zariski closed subgroup) has aLie algebra which is naturally contained in gl(g). The algebraic adjoint group of g is thesmallest algebraic subgroup G ⊆ GL(g) whose Lie algebra contains ad g = {adx | x ∈ g}(cf. [2, §12.2; 50, Definition 24.8.1]).

The natural action of GL(g) on g by linear automorphisms restricts to an action of Gon g, the adjoint action. This, in turn, induces a (left) action of G on g∗, the coadjoint

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16 K. R. GOODEARL

action, under which(g.α)(x) = α(g−1.x)

for g ∈ G, α ∈ g∗, and x ∈ g. The orbits of this action, the coadjoint orbits , are collectedin the set g∗/G. We equip g∗/G with the quotient topology induced from the Zariskitopology on g∗, and thus refer to it as the space of coadjoint orbits.

5.2. Prime and primitive spectra. For any algebra A, we denote the collection of allprimitive ideals of A by primA. This set supports a Zariski topology, under which theclosed sets are the sets V (I) := {P ∈ primA | P ⊇ I} for ideals I of A. We treat primA asa topological space with this topology, and refer to it as the primitive spectrum of A. Theanalogous process, applied to the set of all prime ideals of A, results in the prime spectrum

of A, denoted specA. Since primitive ideals are prime, primA ⊆ specA. In fact, primAis a subspace of specA, that is, its topology coincides with the relative topology inheritedfrom specA. Finally, we shall need the subspace of specA consisting of all the maximalideals of A. This is the maximal ideal space of A, denoted maxspecA.

5.3. The Dixmier map. Let g be a solvable finite dimensional complex Lie algebra.Following [2, §10.8], we use the name Dixmier map and the label Dx for the map

Dx : g∗ −→ primU(g)

introduced by Dixmier in [9]. We do not give the definition here, but just refer to [2]. Itturns out that this map is constant on G-orbits, and so it induces a factorized Dixmier

map

Dx : g∗/G −→ primU(g)

[2, §12.4]. Work of Dixmier, Conze, Duflo, and Rentschler led to the result that Dx is acontinuous bijection [2, Satze 13.4, 15.1]. The conjecture that it is a homeomorphism wasestablished later by Mathieu [35, Theorem], resulting in the following theorem:

5.4. Theorem. [Dixmier-Conze-Duflo-Rentschler-Mathieu] Let g be a solvable finite di-

mensional complex Lie algebra, and G its adjoint algebraic group. Then the factorized

Dixmier map Dx is a homeomorphism from g∗/G onto primU(g).

5.5. Algebraic versus non-algebraic cases. If g is an algebraic Lie algebra, meaningthat it is the Lie algebra of some algebraic group, then the adjoint algebraic group G isa Lie group, and its coadjoint orbits in g∗ are the symplectic leaves for the KKS Poissonstructure, by Theorem 4.2. Otherwise, G is larger than the relevant Lie group, in the sensethat its Lie algebra properly contains ad g. In this case, its coadjoint orbits are largertoo, typically larger than individual symplectic leaves. Our basic example illustrates thisbehavior.

5.6. Example. Return to Example 3.8, and again place the x1x2x3-coordinates of pointsof g∗ in column vectors. The adjoint algebraic group G, written so as to act by leftmultiplication on column vectors, can be expressed as

G =

1 u v0 t 00 0 t′

∣∣∣∣ u, v ∈ C, t, t′ ∈ C×

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 17

[50, §24.8.4]. The coadjoint orbits of G in g∗ are the following sets:

• the individual points on the x1-axis;• the x1x2-plane minus the x1-axis;• the x1x3-plane minus the x1-axis;• g∗ minus the x1x2- and x1x3-planes.

Comparing with Example 3.8, we see that the first three G-orbits are symplectic leaves,while the fourth is not. However, the fourth is at least a union of symplectic leaves.

The fourth G-orbit above is Zariski dense in g∗, while the others are not. Viewingthese orbits as points in the orbit space g∗/G, we find that g∗/G has a unique dense point(i.e., a unique dense singleton subset). By Theorem 5.4, the same holds for primU(g).(Translated into ideal theory, this means that there is one primitive ideal of U(g) which iscontained in all other primitive ideals.) On the other hand, all the surfaces (x3 = λxα2 6= 0)are Zariski dense in g∗, and so the quotient topology on the space of symplectic leaves ing∗ has uncountably many dense points. Therefore this space of symplectic leaves cannotbe homeomorphic to primU(g).

6. Poisson ideal theory and symplectic cores

Since the concept of symplectic leaves is differential-geometric, it should not be sosurprising that it is not always suited to describe answers to algebraic problems, as seen inthe previous section. Consequently, we look for an algebraic replacement. This is providedby Brown and Gordon’s notion of symplectic cores , which is described via the ideal theoryof Poisson algebras.

6.1. Poisson prime ideals. Let R be a (commutative) Poisson algebra (recall §3.1).A Poisson ideal of R is any ideal I of the ring R which is also a Lie ideal relative to

{−,−}, that is, {R, I} ⊆ I. Sums, products, and intersections of Poisson ideals are againPoisson ideals. Whenever I is a Poisson ideal of R, the Poisson bracket on R induces awell defined Poisson bracket on R/I, so that R/I becomes a Poisson algebra.

The Poisson core of an arbitrary ideal J of R is the largest Poisson ideal contained inJ . This exists and is unique, because it is the sum of all Poisson ideals contained in J . Weuse P(J) to denote the Poisson core of J .

A Poisson-prime ideal of R is any proper Poisson ideal P of R with the followingproperty: whenever the product of Poisson ideals I and J of R is contained in P , one of Ior J must be contained in P . Obviously any prime Poisson ideal is Poisson-prime, but theconverse can fail in positive characteristic. As we shall see in a moment, (Poisson-prime)is the same as (prime Poisson) when R is noetherian and k has characteristic zero; in thatcase, we will drop the hyphen and speak of Poisson prime ideals . Note also that if Q is anarbitrary prime ideal of R, then P(Q) is a Poisson-prime ideal.

The Poisson-prime spectrum of R, denoted P.specR, is the set of all Poisson-primeideals of R, equipped with the natural Zariski-type topology, in which the closed sets arethose of the form VP (I) := {P ∈ P.specR | P ⊇ I}, for ideals I of R. It suffices to considerPoisson ideals in defining closed sets, since the ideal I in the definition of a closed set can

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18 K. R. GOODEARL

be replaced by the Poisson ideal it generates. (This observation is helpful in showing thatfinite unions of closed sets are closed.)

6.2. Lemma. Let R be a Poisson k-algebra, where char k = 0. Then the Poisson core of

every prime ideal of R is prime, and all minimal prime ideals of R are Poisson ideals. If

R is noetherian, the Poisson-prime ideals of R coincide with the prime Poisson ideals.

Proof. Commutativity is not needed for this result. The commutative case is covered, forinstance, by [13, Lemma 1.1], and the general case is proved the same way. We sketch thedetails for the reader’s convenience.

The first conclusion is a consequence of [10, Lemma 3.3.2], and the second follows.Now assume that R is noetherian, and let P be a Poisson-prime ideal of R. There exist

prime ideals Q1, . . . , Qt minimal over P such that Q1Q2 · · ·Qt ⊆ P . The minimal primeideals Qi/P in the Poisson algebra R/P must be Poisson ideals by what has been provedso far, and hence the Qi are Poisson ideals of R. Poisson-primeness of P then implies thatsome Qj ⊆ P , whence P = Qj , proving that P is prime. �

6.3. Poisson-primitive ideals and symplectic cores. Let R be a (commutative) Pois-son algebra.

The Poisson-primitive ideals of R are the Poisson cores of the maximal ideals of R.Note from §6.1 that all Poisson-primitive ideals are Poisson-prime.

This terminology is chosen to reflect the following parallel. An ideal P in an algebra Ais left primitive if and only if P is the largest ideal contained in some maximal left ideal.If we view A as a (noncommutative) Poisson algebra via the commutator bracket [−,−],then the ideals of A are precisely the Poisson left ideals. Thus, the left primitive ideals ofA are exactly the Poisson cores of the maximal left ideals.

The Poisson-primitive spectrum of R, denoted P.primR, is the set of all Poisson-primitive ideals of R. This is a subset of P.specR, and we give it the relative topology.

By definition, the process of taking Poisson cores defines a surjective map

maxspecR −→ P.primR,

and we note that this map is continuous. Its fibres, namely the sets

{m ∈ maxspecR | P(m) = P}

for P ∈ P.primR, are called symplectic cores . They determine a partition of maxspecR.Now suppose that R = O(V ) is the coordinate ring of an affine variety V , and that k

is algebraically closed. As in the complex case, we say that V is a Poisson variety . Sincek is algebraically closed, there is a natural identification V ≡ maxspecR, with which wetransfer the symplectic cores from maxspecR to V . In other words, the symplectic cores

in V are the sets{p ∈ V | P(mp) = P}

for P ∈ P.primR, where mp = {f ∈ R | f(p) = 0}.

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 19

6.4. Example. Return to Example 3.8, and set R = O(g∗) = C[x1, x2, x3]. The Poisson-primitive ideals of R can be computed as follows:

P(〈x1 − α, x2, x3〉) = 〈x1 − α, x2, x3〉 (α ∈ C)

P(〈x1 − α, x2 − β, x3〉) = 〈x3〉 (α ∈ C, β ∈ C×)

P(〈x1 − α, x2, x3 − γ〉) = 〈x2〉 (α ∈ C, γ ∈ C×)

P(〈x1 − α, x2 − β, x3 − γ〉) = 〈0〉 (α ∈ C, β, γ ∈ C×).

It follows that the symplectic cores in g∗ are the sets

• the individual points on the x1-axis;• the x1x2-plane minus the x1-axis;• the x1x3-plane minus the x1-axis;• g∗ minus the x1x2- and x1x3-planes.

These are precisely the coadjoint orbits of the adjoint algebraic group of g, as we saw inExample 5.6.

7. Symplectic cores versus symplectic leaves

Symplectic cores are related to symplectic leaves by the following result of Brown andGordon [6, Lemma 3.3 and Proposition 3.6]; further relations will be given below. Here“locally closed” refers to the Zariski topology.

7.1. Theorem. [Brown-Gordon] Let V be a complex affine Poisson variety.

(a) Each symplectic core in V is locally closed, and is a union of symplectic leaves.

(b) If the symplectic leaves in V are all locally closed, then they coincide with the sym-

plectic cores.

It is a standard result that the orbits of a connected algebraic group G acting on avariety X can be recovered from the orbit closures, as follows. Take any orbit closure C,and remove all orbit closures properly contained in C. The result will be a single G-orbit,and all G-orbits in X are obtained by this means. Yakimov has conjectured that thesymplectic cores in a complex affine Poisson variety can be recovered from the closuresof the symplectic leaves in a similar manner. We verify this below, with the help of thefollowing lemma of Brown and Gordon [6, Lemma 3.5]. All topological properties are tobe taken relative to the Zariski topology.

7.2. Lemma. [Brown-Gordon] Let V be a complex affine Poisson variety, and R = O(V ).Let L be a symplectic leaf in V , and set K = {f ∈ R | f = 0 on L}. Then K is a Poisson-

primitive ideal of R, and L is contained in the corresponding symplectic core, that is,

P(mp) = K for all p ∈ L.7.3. Lemma. Let V be a complex affine Poisson variety, and R = O(V ). Let K be a

Poisson ideal of R, and X the closed subvariety of V determined by K. Then X is a

union of symplectic cores and a union of symplectic leaves. In particular, the closure of

any symplectic leaf of V is a union of symplectic leaves.

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20 K. R. GOODEARL

Proof. If p ∈ X , then mp ⊇ K. Since K is a Poisson ideal, it must be contained inthe Poisson-primitive ideal P = P(mp). Now the set C = {q ∈ V | P(mq) = P} is thesymplectic core containing p, and C ⊆ X because mq ⊇ P ⊇ K for all q ∈ C. ThereforeX is a union of symplectic cores. That X is a union of symplectic leaves now follows fromTheorem 7.1(a).

For any symplectic leaf L of V , the ideal I of functions in R that vanish on L is aPoisson ideal by Lemma 7.2. The closed subvariety determined by I is the closure of L,and this is a union of symplectic leaves by what we have just proved. �

We can now prove that symplectic cores are obtained from symplectic leaves in themanner proposed by Yakimov; this is parts (c) and (e) of the following theorem. Hereoverbars denote closures.

7.4. Theorem. Let V be a complex affine Poisson variety, and L a symplectic leaf in V .

(a) There is a unique symplectic core C in V containing L, and C ⊆ L.(b) C is the union of those symplectic leaves of V which are dense in L.

(c) C = L \⋃

M M where M runs over those symplectic leaves whose closures are

properly contained in L.

(d) C is the unique symplectic core dense in L.

(e) Each symplectic core in V is dense in the closure of every symplectic leaf it contains.

Hence, it can be obtained from the closure of such a leaf as in part (c).

Proof. Set R = O(V ), and let K be the ideal of functions in R that vanish on L.

(a) The symplectic cores and the symplectic leaves both partition V , and the latter forma finer partition, by Theorem 7.1(a). This implies the existence and uniqueness of C.

By Lemma 7.2, K is a Poisson-primitive ideal, and the symplectic core it determinescontains L. By uniqueness, this core is C, that is, C = {p ∈ V | P(mp) = K}. In

particular, mp ⊇ K for all p ∈ C, from which it follows that C ⊆ L.(b) IfM is a symplectic leaf which is dense in L, then K equals the ideal of functions in

R that vanish on M , and Lemma 7.2 implies that M ⊆ C. On the other hand, if M ′ is asymplectic leaf which is contained in but not dense in L, the ideal K ′ of functions vanishingon M ′ properly contains K, whence M ′ is contained in a symplectic core different fromC. In this case, M ′ is disjoint from C. Part (b) now follows, because L is a union ofsymplectic leaves, by Lemma 7.3.

(c) In view of Lemma 7.3, the given union⋃

M M equals the union of those symplectic

leaves which are contained in L but not dense in L. The given formula for C thus followsfrom part (b).

(d) Clearly C is dense in L, since L ⊆ C ⊆ L. If D is a different symplectic corecontained in L, then by (b), any symplectic leaf N ⊆ D is not dense in L. But D ⊆ N by(a), and thus D is not dense in L.

(e) Suppose that D is a symplectic core in V , and N a symplectic leaf contained in D.By (a), D is the unique symplectic core containing N , and D ⊆ N , whence D is dense inN . The final statement now follows from (c), with C and L replaced by D and N . �

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 21

8. Symplectic cores versus primitive ideals for solvable Lie algebras

We now show that the concept of symplectic cores exactly overcomes the limitations ofsymplectic leaves with respect to the Dixmier map discussed in Section 5. Namely, theDixmier map provides a homeomorphism from the space of symplectic cores in g∗ ontothe primitive spectrum of U(g), for any solvable finite dimensional complex Lie algebra g.This just amounts to showing that the coadjoint orbits in g∗, with respect to the adjointalgebraic group of g, coincide with the symplectic cores. Solvability is not needed for thelatter result.

All that is required to obtain the new statement about the Dixmier map is to reinterpretparts of the development of Theorem 5.4 in terms of the new concepts. This reinterpre-tation also shows that (for g solvable) P.primO(g∗) is homeomorphic to primU(g). Witha little extra effort, we can handle prime ideals as well, showing that P.specO(g∗) ishomeomorphic to specU(g).

Throughout this section, g will denote a finite dimensional complex Lie algebra and Gits adjoint algebraic group. We do not assume g solvable until Theorem 8.5. Some of theresults we will need are developed in the literature in terms of S(g) rather than O(g∗).This requires use of the Poisson isomorphism θ : S(g)

∼=−−→ O(g∗) of (2.6).8.1. Actions of G and g. The group G acts on g and g∗ by the adjoint and coadjointactions, respectively, as in §5.1. In turn, these induce actions of G by C-algebra automor-phisms on S(g) and O(g∗), actions which we also refer to as adjoint and coadjoint actions .All G-actions we mention will refer to one of these four cases. Let us write specG S(g) andspecGO(g∗) for the sets of G-stable prime ideals in S(g) and O(g∗), respectively, equippedwith the relative topologies from specS(g) and specO(g∗).

We claim that the isomorphism θ is G-equivariant. To see this, let {e1, . . . , en} be abasis for g and {α1, . . . , αn} the corresponding dual basis for g∗. As in §2.6, O(g∗) =C[x1, . . . , xn] where each xi = θ(ei). Given γ ∈ G, there are scalars γij ∈ C such thatγ.ej =

∑i γijei for all j. Consequently,

(γ.xj)(αi) = xj(γ−1.αi) = (γ−1.αi)(ej) = αi(γ.ej) = γij

for all i, j, from which we conclude that γ.xj =∑

i γijxi for all j. Therefore γ.θ(ej) =θ(γ.ej) for all j, and the G-equivariance of θ follows.

For each e ∈ g, the Lie derivation ad e = [e,−] on g extends uniquely to a derivation onS(g), namely the Hamiltonian {e,−}. This yields an action of g on S(g) by derivations.We write specg S(g) for the set of g-stable prime ideals of S(g), equipped with the relativetopology from specS(g).

8.2. Lemma. (a) specG S(g) = specg S(g) = P.specS(g).(b) specGO(g∗) = P.specO(g∗).(c) θ induces a homeomorphism specg S(g)

≈−−→ P.specO(g∗).Proof. (a) Since g generates the algebra S(g), the g-stable ideals of S(g) coincide withthe Poisson ideals. Hence, specg S(g) = P.specS(g). By [2, §13.1] or [50, §24.8.3], the

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22 K. R. GOODEARL

g-stable ideals of S(g) coincide with the G-stable ideals. From this, we immediately obtainspecG S(g) = specg S(g).

(b)(c) These follow immediately from (a), because θ is both G-equivariant and a Poissonisomorphism. �

Following our previous notation for maximal ideals corresponding to points in varieties,write mα for the maximal ideal of O(g∗) corresponding to a point α ∈ g∗.

8.3. Proposition. Let g be a finite dimensional complex Lie algebra and G its adjoint

algebraic group. There is a homeomorphism φ : g∗/G→ P.primO(g∗) such that φ(G.α) =P(mα) for all α ∈ g∗.

Proof. Since S(g) is isomorphic to O(g∗), its maximal ideal space is homeomorphic to g∗.A coordinate-free way to express the inverse isomorphism is to send each α ∈ g∗ to theideal mα = 〈e− α(e) | e ∈ g〉 of S(g). Observe that θ(mα) = mα.

By [2, Lemma 13.2 and proof], there is a topological embedding

τ : g∗/G −→ specg S(g)

such that τ(G.α) =⋂

γ∈G γ.mα for α ∈ g∗. Thus, τ(G.α) is the largest G-stable ideal

of S(g) contained in mα. Invoking [2, §13.1] or [50, §24.8.3] again, we find that τ(G.α)is the largest g-stable ideal of S(g) contained in mα. In particular, it now follows from[10, Lemma 3.3.2] that τ(G.α) is a prime ideal. Hence, we can say that τ(G.α) equals thelargest member of specG S(g) contained in mα. Since θ is G-equivariant, it follows thatθτ(G.α) equals the largest member of specGO(g∗) contained in mα. In view of Lemma8.2(b), we conclude that θτ(G.α) = P(mα).

Combining the above with Lemma 8.2(c), we obtain a topological embedding

φ : g∗/G→ P.specO(g∗)

such that φ(G.α) = P(mα) for α ∈ g∗. Since the image of φ is, by definition, P.primO(g∗),the proposition is proved. �

8.4. Corollary. Let g be a finite dimensional complex Lie algebra and G its adjoint alge-

braic group. The G-orbits in g∗ are precisely the symplectic cores.

Proof. Injectivity and well-definedness of the homeomorphism φ of Proposition 8.3 saythat for all α, β ∈ g∗, we have G.α = G.β if and only if P(mα) = P(mβ). Thus, α and βlie in the same G-orbit if and only if they lie in the same symplectic core. �

Corollary 8.4 allows us to phrase the Dixmier-Conze-Duflo-Rentschler-Mathieu Theoremin terms of symplectic cores:

8.5. Theorem. Let g be a solvable finite dimensional complex Lie algebra, and let X be

the set of symplectic cores in g∗, with the quotient topology induced from g∗. Then the

Dixmier map induces a homeomorphism X → primU(g). �

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 23

8.6. The extended Dixmier map. Continue to assume that g is solvable. Via theembedding g∗/G −→ specg S(g) from [2, Lemma 13.2] used above, identify g∗/G with asubspace of specg S(g). Borho, Gabriel, and Rentschler showed that Dx extends uniquelyto a continuous map

Dx : specg S(g)→ specU(g),

given by the rule

Dx(P ) =⋂{Dx(α) | α ∈ g∗ and mα ⊇ P}

for P ∈ specg S(g) [2, Satz 13.4]. They named this the extended Dixmier map, and provedthat it is a continuous bijection [2, Satz 13.4, Kor. 15.1]. Their methods, combined with

Mathieu’s theorem, imply that Dx is a homeomorphism, as we will see shortly.

8.7. Quasi-homeomorphisms and sauber spaces. Let X and Y be topological spaces.A continuous map φ : X → Y is a quasi-homeomorphism provided the induced mapF 7→ φ−1(F ) is an isomorphism from the lattice of closed subsets of Y onto the latticeof closed subsets of X . If X is a subspace of Y , the inclusion map X → Y is a quasi-homeomorphism if and only if F ∩X = F for all closed sets F ⊆ Y [2, §1.6]. Borho,Gabriel, and Rentschler observed that the inclusion map primU(g) → specU(g) is aquasi-homeomorphism [2, Beispiel 1.6], as is the above embedding g∗/G −→ specg S(g) [2,Lemma 13.2].

A generic point of a closed subset F ⊆ X is any point x ∈ F such that F = {x}. ThespaceX is sauber (English: tidy) provided every irreducible closed subset ofX has preciselyone generic point. As observed in [2, §13.3], the prime spectrum of any noetherian ring issauber. We include the short argument in the lemma below, for the reader’s convenience.The same argument shows that specg S(g) is sauber. These spaces are noetherian as well,since they have Zariski topologies arising from noetherian rings.

8.8. Lemma. Let A be a noetherian ring and R a commutative noetherian Poisson k-al-gebra, with char k = 0.

(a) The prime spectrum specA is a sauber noetherian space, and if A is a Jacobson

ring, the inclusion map primA→ specA is a quasi-homeomorphism.

(b) The Poisson prime spectrum P.specR is a sauber noetherian space, and if R is an

affine k-algebra, the inclusion map P.primR→ P.specR is a quasi-homeomorphism.

Proof. (a) Suppose that F1 ⊇ F2 ⊇ · · · is a decreasing sequence of closed sets in specA.We may write each Fj = V (Ij) where Ij =

⋂Fj . Then I1 ⊆ I2 ⊆ · · · is an increasing

sequence of ideals of A. Since this sequence stabilizes, so does the original sequence ofclosed sets. Thus, specA is a noetherian space.

Let F = V (I) be an arbitrary closed subset of specA, where I is an ideal of A. Wemay replace I by its prime radical, so there is no loss of generality in assuming that I issemiprime. Since A is noetherian, there are only finitely many prime ideals minimal overI, say Q1, . . . , Qn, and I = Q1 ∩ · · · ∩Qn. It follows that F = V (Q1) ∪ · · · ∪ V (Qn).

If F is irreducible, then F = V (Qj) for some j. In this case, Qj is the unique genericpoint of F , proving that specA is sauber.

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24 K. R. GOODEARL

Now assume that A is a Jacobson ring, so that all prime ideals of A are intersections ofprimitive ideals. It follows that

I =⋂

F =⋂

(F ∩ primA),

from which we see that F equals the closure of F ∩ primA in specA. Thus, by [2, §1.6],the inclusion map primA→ specA is a quasi-homeomorphism.

(b) The argument applied in (a) also shows that P.specR is a noetherian space.As discussed in §6.1, any closed set F in P.specR can be written F = VP (I) for some

Poisson ideal I. There are only finitely many prime ideals minimal over I, say Q1, . . . , Qn,and the Qi are Poisson ideals by Lemma 6.2. Hence, we may replace I by Q1 ∩ · · · ∩Qn,and it follows that F = VP (Q1) ∪ · · · ∪ VP (Qn).

Just as in (a), if F is irreducible, F = VP (Qj) for some j, and then Qj is the uniquegeneric point of F . This proves that P.specR is sauber.

Now assume that R is an affine k-algebra. Then R is a Jacobson ring, and it followsthat every Poisson prime ideal of R is an intersection of Poisson-primitive ideals (e.g., see[13, Lemma 1.1(e)]). From this, we conclude as in (a) that the inclusion map P.primR→P.specR is a quasi-homeomorphism. �

8.9. Lemma. Let X ⊆ X ′ and Y ⊆ Y ′ be topological spaces, such that X ′ and Y ′ are

sauber and noetherian. Assume also that the inclusion maps X → X ′ and Y → Y ′ are

quasi-homeomorphisms. Then any continuous map φ : X → Y extends uniquely to a

continuous map φ′ : X ′ → Y ′. Moreover, if φ is a homeomorphism, so is φ′.

Proof. The existence and uniqueness of φ are proved in [2, Lemma 13.3]. The final state-ment follows by the usual universal property argument. �

8.10. Theorem. [Borho-Gabriel-Rentschler-Mathieu] Let g be a solvable finite dimen-

sional complex Lie algebra. The extended Dixmier map

Dx : specg S(g) −→ specU(g)

is a homeomorphism.

Proof. Following the proof of [2, Satz 13.4], recall that specg S(g) and specU(g) aresauber noetherian spaces, and that the embedding g∗/G → specg S(g) and the inclusion

primU(g)→ specU(g) are quasi-homeomorphisms. The map Dx is defined, with the helpof Lemma 8.9, to be the unique continuous map from specg S(g) to specU(g) extending

Dx. Since Dx is a homeomorphism, Lemma 8.9 implies that Dx is a homeomorphism. �

In Poisson-ideal-theoretic terms, Theorems 5.4 and 8.10 can be restated as follows.

8.11. Theorem. Let g be a solvable finite dimensional complex Lie algebra. Then there

is a homeomorphism

ψ : P.primO(g∗) −→ primU(g)

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 25

such that ψ(P(mα)) = Dx(α) for α ∈ g∗, and ψ extends uniquely to a homeomorphism

P.specO(g∗) −→ specU(g).

Proof. To obtain ψ, just compose the factorized Dixmier map Dx with the inverse ofthe homeomorphism φ of Proposition 8.3. By Lemma 8.8, P.specO(g∗) and specU(g)are sauber noetherian spaces, and the inclusion maps P.primO(g∗) → P.specO(g∗) andprimU(g) → specU(g) are quasi-homeomorphisms. Therefore the existence and unique-ness of the desired extension of ψ follow from Lemma 8.9. �

9. Modified conjectures for quantized coordinate rings

In light of Theorems 7.1, 7.4, 8.5, and 8.11, we nominate the concept of symplectic coresas the best algebraic approximation for symplectic leaves. Further, we suggest that sym-plectic leaves should be replaced by symplectic cores in applications of the Orbit Methodto algebraic problems. In particular, we revise and refine the general principle discussedin §4.4 to the following conjecture. It is, of necessity, somewhat imprecise, given the lackof a precise definition of the concept of quantized coordinate rings.

9.1. Primitive spectrum conjecture for quantized coordinate rings.

Assume that k is algebraically closed of characteristic zero, and let A be a genericquantized coordinate ring of an affine algebraic variety V over k. Then A shouldbe a member of a flat family of k-algebras with semiclassical limit O(V ), suchthat primA is homeomorphic to the space of symplectic cores in V , with respectto the semiclassical limit Poisson structure. Further, there should be compatiblehomeomorphisms primA→ P.primO(V ) and specA→ P.specO(V ).

Each of the known types of quantized coordinate rings supports an action of an algebraictorus H = (k×)m (see [4, §§II.1.14-18] for a summary), which has a parallel action (byPoisson automorphisms) on the semiclassical limit (e.g., see [19, §0.2; 14, Section 2]).We tighten the conjecture above and posit that there should exist homeomorphisms asdesribed which are also equivariant with respect to the relevant torus actions.

9.2. Remarks. (a) The discussion of the simple example Aq = Oq(C2) in §4.5 indicates

why Conjecture 9.1 is restricted to generic quantized coordinate rings. In particular,primAq has a generic point when q is not a root of unity, but no generic points otherwise.Since P.specC[x, y] has a generic point, it is not homeomorphic to primAq when q is aroot of unity.

(b) Each of the “standard” single parameter quantized coordinate rings is defined as amember of a one-parameter family of algebras, and it is this (flat) family to which the con-jecture is meant to apply. For instance, the algebras Oq(SLn(k)) (with n fixed) are definedfor all q ∈ k× in the same way (e.g., [4, §I.2.4]), and substituting an indeterminate t for qin the definition results in a torsionfree k[t±1]-algebra A with A/(t − q)A ∼= Oq(SLn(k))for all q ∈ k×, just as with the case n = 2 in §§1.6, 2.2(c). The semiclassical limit is

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26 K. R. GOODEARL

O(SLn(k)) with the Poisson bracket satisfying

{Xij, Xim} = XijXim (j < m)

{Xij , Xlj} = XijXlj (i < l)

{Xij , Xlm} = 0 (i < l, j > m)

{Xij , Xlm} = 2XimXlj (i < l, j < m).

This Poisson structure and the above flat family should feature in the SLn case of Conjec-ture 9.1, that is, for q not a root of unity, primOq(SLn(k)) should be homeomorphic to thespace of symplectic cores in SLn(k) and to P.primO(SLn(k)), and specOq(SLn(k)) shouldbe homeomorphic to P.specO(SLn(k)). Such a “standard” version of the conjecture is tobe posed for Oq(Mn(k)), Oq(GLn(k)), Oq(G), and other “standard” cases.

The situation is more involved for “nonstandard” cases, and for multiparameter families,which have to be reduced to single parameter families in order to obtain semiclassical limits.In such cases, the conjecture may be sensitive to the choice of flat family – different flatfamilies may yield different Poisson structures in the semiclassical limit, and the conjecturemay hold for some of these semiclassical limits but not for others. This phenomenonappears in an example of Vancliff [52, Example 3.14], which we discuss in Example 9.9.

(c) As discussed at the end of Example 2.6, the enveloping algebra of a finite dimensionalLie algebra g is a generic member of the flat family given by the homogenization of U(g),and so U(g) should qualify as a generic quantized coordinate ring of g∗. The semiclassicallimit of this family is the Poisson algebra O(g∗). For this setting, K. A. Brown has noteddifficulties with Conjecture 9.1 in what one might expect to be the most canonical case,namely when g is semisimple [3]. Following the Orbit Method, one would seek a bijectionL ←→ P between symplectic leaves in g∗ and primitive ideals in U(g) such that theGelfand-Kirillov dimension of U(g)/P equals the dimension of L. In particular, the zero-dimensional symplectic leaves of g∗, which are the same as the zero-dimensional symplecticcores, should match up with the maximal ideals of finite codimension in U(g). However,U(g) has infinitely many such maximal ideals, while there is only one zero-dimensionalsymplectic leaf in g∗. (The latter can be verified by using Theorem 4.2 together with thefact that the identification of g∗ with g via the Killing form identifies the coadjoint orbitsin g∗ with the adjoint orbits in g [8, p. 12].)

Other differences are already visible in the case g = sl2(C). As is easily computed, allbut one of the coadjoint orbits in g∗ are closed (compare with the adjoint orbits, computedin [8, Example 1.2.1]). It follows (using Proposition 8.3, or by direct computation) that allbut one of the points of P.primO(g∗) are closed. However, primU(g) has infinitely manynon-closed points, and therefore it is not homeomorphic to P.primO(g∗).

(d) Whenever Conjecture 9.1 does hold, the space of symplectic cores in V must behomeomorphic to P.primO(V ). It is an open question whether the space of symplecticcores for an arbitrary affine Poisson algebra R is homeomorphic to P.primR, but thisdoes hold when R satisfies the Poisson Dixmier-Moeglin equivalence, as follows from [13,Theorem 1.5]; we excerpt the basic argument in Lemma 9.3. This equivalence requiresthat the Poisson-primitive ideals of R coincide with the locally closed points of P.specR,

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 27

and with those Poisson prime ideals P of R for which the Poisson center (cf. §9.6(b)) ofthe quotient field of R/P is algebraic over k. It holds for the semiclassical limits of manyquantized coordinate rings via [13, Theorem 4.1], as shown in [14, Section 2].

(e) As in Theorem 8.11, the existence of a homeomorphism primA→ P.primO(V ) as inthe conjecture typically implies the existence of a compatible homeomorphism specA →P.specO(V ). We display this in Lemma 9.4 below for emphasis. On the other hand,a homeomorphism specA → P.specO(V ) will restrict to a homeomorphism primA →P.primO(V ) provided O(V ) satisfies the Poisson Dixmier-Moeglin equivalence and A sat-isfies the Dixmier-Moeglin equivalence. The latter equivalence requires that the primitiveideals of A coincide with the locally closed points of specA, and with those prime idealsP of A for which Z(FractA/P ) is algebraic over k. It was verified for many quantizedcoordinate rings in [16] (see [4, Corollary II.8.5] for a summary).

9.3. Lemma. Let R be a commutative affine Poisson k-algebra, and assume that all

Poisson-primitive ideals of R are locally closed points in P.specR. Then the Zariski topol-

ogy on P.primR coincides with the quotient topology induced by the Poisson core map

P(−) : maxspecR→ P.primR. Consequently, the space of symplectic cores in maxspecRis homeomorphic to P.primR.

Proof. Observe first that the map P(−) is continuous. It is surjective by definition ofP.primR.

We claim that P =⋂{m ∈ maxspecR | P(m) = P}, for any Poisson-primitive ideal P

of R. Since P is locally closed in P.specR (by assumption), the singleton {P} is open inits closure VP (P ), and so {P} = VP (P ) \ VP (J) for some Poisson ideal J of R. Note thatJ 6⊆ P ; hence, after replacing J by J + P , we may assume that J ) P . If m ⊇ P is amaximal ideal such that P(m) 6= P , then m ⊇ P(m) ⊇ J . The remaining maximal idealscontaining P must intersect to P by the Nullstellensatz, verifying the claim.

Now consider a set X ⊆ P.primR whose inverse image under P(−), call it Y , is closedin maxspecR. Thus,

Y = {m ∈ maxspecR | P(m) ∈ X} = {m ∈ maxspecR | m ⊇ I}

for some ideal I of R. If P ∈ X and m ∈ maxspecR with P(m) = P , then m ∈ Y , and som ⊇ I. By the claim above, the intersection of these maximal ideals equals P , and thusP ⊇ I. Conversely, if P ∈ P.primR and P ⊇ I, then P = P(m) for some maximal idealm ⊇ I, whence m ∈ Y and P ∈ X . Therefore X = {P ∈ P.primR | P ⊇ I}, a closed setin P.primR. This proves that the topology on P.primR is the quotient topology inheritedfrom maxspecR via P(−).

The final statement of the lemma follows directly. �

9.4. Lemma. Let A be a noetherian k-algebra and R a commutative noetherian Poisson

k-algebra, with char k = 0.(a) A bijection φ : specA → P.specR is a homeomorphism if and only if φ and φ−1

preserve inclusions.

(b) Assume that A is a Jacobson ring and R an affine k-algebra. Then any homeomor-

phism primA→ P.primR extends uniquely to a homeomorphism specA→ P.specR.

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28 K. R. GOODEARL

(c) Assume that A satisfies the Dixmier-Moeglin equivalence and R the Poisson Dixmier-

Moeglin equivalence. Then any homeomorphism specA → P.specR restricts to a homeo-

morphism primA→ P.primR.

Proof. (a) For P,Q ∈ specA, we have P ⊆ Q if and only if Q ∈ {P}, and similarly inP.specR. Hence, any homeomorphism between these spaces must preserve inclusions.

Conversely, if φ and φ−1 preserve inclusions, then φ(V (P )) = VP (φ(P )) for all P ∈specA. Since the closed sets in specA are exactly the finite unions of V (P ) s (recall theproof of Lemma 8.8(a)), it follows that φ sends closed sets to closed sets, i.e., φ−1 iscontinuous. Similarly, φ is continuous, and hence a homeomorphism.

(b) Lemmas 8.8 and 8.9.

(c) Under the assumed equivalences, primA consists of the locally closed points inspecA, and P.primR consists of the locally closed points in P.specR. �

9.5. Example. Let Aq = Oq(k2), where k = k, char k = 0, and q ∈ k×. View R = O(k2)

as the semiclassical limit of the family (Aq)q∈k×, with the Poisson structure exhibited inExample 2.2(a). The torus H = (k×)2 acts on Aq via algebra automorphisms and on Rvia Poisson automorphisms so that (in both cases) (α1, α2).xi = αixi for (α1, α2) ∈ H andi = 1, 2.

Assume that q is not a root of unity. As is easily checked (e.g., [4, Example II.1.2]), theprime ideals of Aq are

• the maximal ideals 〈x1 − α, x2〉 and 〈x1, x2 − β〉, for α, β ∈ k;(♦) • the height 1 primes 〈x1〉 and 〈x2〉;• the zero ideal.

All of these prime ideals, except for 〈x1〉 and 〈x2〉, are primitive [4, Example II.7.2]. Theclosed sets in specAq are easily found, but we shall not list them here – see [4, ExampleII.1.2 and Exercise II.1.C].

With very similar computations, one finds the Poisson prime and Poisson-primitiveideals in R, and a list of the closed subsets of P.specR. In terms of notation, the answersare the same as for Aq – the list (♦) also describes the Poisson prime ideals of R, and allof these ideals, except for 〈x1〉 and 〈x2〉, are Poisson-primitive. We conclude that thereexist compatible homeomorphisms primAq → P.primR and specAq → P.specR, sendingthe ideal 〈x1 − α, x2〉 of Aq to the ideal 〈x1 − α, x2〉 of R, and so on. (We say that thesemaps are given by “preservation of notation”.) These homeomorphisms are equivariantwith respect to the actions of H described above.

By inspection, all Poisson-primitive ideals of R are locally closed in P.specR. Conse-quently, we conclude from Lemma 9.3 that the space of symplectic cores in maxspecR ≈ k2is homeomorphic to P.primR.

Analyzing the prime ideals in a quantized coordinate ring typically involves investigatinglocalizations of factor algebras, which often turn out to be quantum tori. We sketch somebasic procedures used to determine prime ideals in quantum tori, and similar ones for theanalogous “Poisson tori”.

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 29

9.6. Some computational tools. (a) A quantum torus over k is the localization of aquantum affine space Oq(k

n) obtained by inverting the generators xi, that is, an algebra

Oq((k×)n) = k〈x±1

1 , . . . , x±1n | xixj = qijxjxi for all i, j〉,

where q = (qij) is a multiplicatively antisymmetric n×nmatrix over k. Set T = Oq((k×)n).

Since T is a Zn-graded algebra, with 1-dimensional homogeneous components, its centeris spanned by central monomials [21, Lemma 1.1]. The latter are easily computed: amonomial xm1

1 xm2

2 · · ·xmnn is central if and only if

∏nj=1 q

mj

ij = 1 for all i. All ideals of

T are induced from ideals of Z(T ) [21, Theorem 1.2; 15, Proposition 1.4], from which itfollows that contraction and extension give inverse homeomorphisms between spec T andspecZ(T ) [15, Corollary 1.5(b)]. In particular, it follows from the above facts that T is asimple algebra if and only if Z(T ) = k [36, Proposition 1.3].

(b) Let R = k[x±11 , . . . , x±1

n ] be a Laurent polynomial ring, equipped with a Poissonbracket such that {xi, xj} = πijxixj for all i, j, where (πij) is an antisymmetric n × nmatrix over k. The results of part (a) all have Poisson analogs for R, as follows.

The Poisson center of R, denoted ZP (R), is the subalgebra consisting of those r ∈ Rfor which the derivation {r,−} vanishes. Since the Poisson bracket on R respects theZn-grading, ZP (R) is spanned by the monomials it contains [21, Lemma 2.1; 52, Lemma1.2(a)]. A monomial xm1

1 xm2

2 · · ·xmnn is Poisson central if and only if

∑nj=1 πijmj = 0 for all

i. All Poisson ideals of R are induced from ideals of ZP (R) [21, Theorem 2.2; 52, Lemma1.2(b)], from which it follows that contraction and extension give inverse homeomorphismsbetween P.specR and specZP (R). In particular, it follows from the above facts that Ris Poisson-simple (meaning that it has no proper nonzero Poisson ideals) if and only ifZP (R) = k.

9.7. Example. Let Aq = Oq(SL2(k)), where k = k, char k = 0, and q ∈ k×. ViewR = O(SL2(k)) as the semiclassical limit of the family (Aq)q∈k×, with the Poisson structureexhibited in Example 2.2(c). The torus H = (k×)2 again acts on Aq and R, this time sothat

(α, β).X11 = αβX11 (α, β).X12 = αβ−1X12

(α, β).X21 = α−1βX21 (α, β).X22 = α−1β−1X22

for (α, β) ∈ H.Now restrict q to a non-root of unity. The prime ideals of Aq can be computed with the

tools of §9.6(a), as outlined in [4, Exercise II.1.D]. For instance, one checks that Aq has alocalization

Aq[X−111 , X

−112 , X

−121 ] ∼= k〈x±1, y±1, z±1 | xy = qyx, xz = qzx, yz = zy〉,

and that the center of the latter algebra is k[(yz−1)±1]. It follows that the prime ideals ofAq not containing X12 or X21 consist of 〈0〉 and 〈X12− λX21〉, for λ ∈ k×. The full list ofprime ideals of Aq is as follows:

• the maximal ideals 〈X11 − λ, X12, X21, X22 − λ−1〉, for λ ∈ k×;

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30 K. R. GOODEARL

(♠) • the ideal 〈X12, X21〉;• the height 1 primes 〈X21〉 and 〈X12 − λX21〉, for λ ∈ k;• the zero ideal.

A diagram of specAq, with inclusions marked, is given in [4, Diagram II.1.3].A similar computation, using §9.6(b), yields the Poisson prime ideals of R, which

can be described exactly as in (♠). This provides a natural H-equivariant bijectionφ : specAq → P.specR, given by “preservation of notation”. By inspection, φ and φ−1

preserve inclusions, and thus, by Lemma 9.4(a), φ is a homeomorphism.The algebra Aq satisfies the Dixmier-Moeglin equivalence by [4, Corollary II.8.5], and R

satisfies the Poisson Dixmier-Moeglin equivalence by [13, Theorem 4.3]. Therefore Lemma9.4(c) implies that φ restricts to a homeomorphism primAq → P.primR. In Aq, all primeideals are primitive except for 〈X12, X21〉 and 〈0〉 (cf. [4, Example II.8.6]). Similarly, inR all Poisson prime ideals are Poisson-primitive except for 〈X12, X21〉 and 〈0〉. As inthe previous example, we can use Lemma 9.3 to see that the space of symplectic cores inmaxspecR ≈ SL2(k) is homeomorphic to P.primR.

9.8. Evidence for Conjecture 9.1. In most of the instances discussed below, k isassumed to be algebraically closed of characteristic zero.

(a) Examples 9.5 and 9.7 are the most basic instances in which the conjecture has beenverified. In the same way (although with somewhat more effort), one can verify it forOq(GL2(k)). In particular, most of the work required to determine the prime ideals in thegeneric Oq(GL2(k)) is done in [4, Example II.8.7].

(b) We next turn to the quantized coordinate rings Oq(G) and Oq,p(G) over k = C,where G is a connected semisimple complex Lie group, q ∈ k× is not a root of unity, andp is an antisymmetric bicharacter on the weight lattice of G (as in [24, §3.4]).

The Poisson structure on O(G) resulting from the semiclassical limit process gives Gthe combined structure of a Poisson-Lie group (e.g., see [33, Chapter 1] for the concept,and [22, §A.1] for the result). There is a known recipe for the symplectic leaves in G incase the Poisson structure arises from the standard quantization [22, Appendix A], andsimilarly in the multiparameter “algebraic” case [24, Theorem 1.8]. In both these cases,it follows that the symplectic leaves are Zariski locally closed (see [5, Theorem 1.9] for amore explicit statement). Hence, the symplectic leaves in G coincide with the symplecticcores (Theorem 7.1(b)).

As discussed in §4.4, Hodges and Levasseur put forward the conjecture that there shouldbe a bijection between primOq(G) and the set of symplectic leaves in G [22, §2.8, Conjec-ture 1]. They developed such bijections for G = SLn(C) in [23], and for general G in theirwork with Toro [24]. More generally, Hodges, Levasseur, and Toro established a bijectionbetween primOq,p(G) and the set of symplectic leaves in G in the algebraic case. All thesebijections are equivariant with respect to natural actions of a maximal torus of G.

Except for the case G = SL2(C) covered in Example 9.7, the topological properties ofthe above bijections are not known. Even when G = SL3(C), it is not known whetherprimOq(G) is homeomorphic to the space of symplectic leaves (= cores) in G.

(c) The prime and primitive spectra of general multiparameter quantum affine spacesOq(k

n) were analyzed by Goodearl and Letzter in [17], assuming k = k together with a mi-

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 31

nor technical assumption (that either char k = 2, or −1 is not in the subgroup 〈qij〉 ⊆ k×).They proved that there are compatible topological quotient maps kn ≈ maxspecO(kn)→primOq(k

n) and specO(kn) → specOq(kn), equivariant with respect to natural actions

of the torus (k×)n [17, Theorem 4.11]. Similar results were proved not only for quantumtori Oq((k

×)n) [17, Theorem 3.11] but also for quantum affine toric varieties [17, Theorem6.3].

Oh, Park, and Shin converted these topological quotient results into the following (as-suming char k = 0 and −1 /∈ 〈qij〉 ⊆ k×): For each Oq(k

n), there is a Poisson structure onO(kn) such that there are compatible homeomorphisms P.primO(kn)→ primOq(k

n) andP.specO(kn) → specOq(k

n) [41, Theorem 3.5]. Goodearl and Letzter, finally, showedthat such homeomorphisms could be obtained for semiclassical limit Poisson structures[18, Theorem 3.6], and extended the results to quantum affine toric varieties [18, Theorem5.2].

All these Poisson algebra structures on O(kn) satisfy the Poisson Dixmier-Moeglinequivalence [13, Example 4.6]. Hence, the space of symplectic cores in kn is homeomorphicto primOq(k

n), via Lemma 9.3. The symplectic cores in kn are algebraic, whereas thisdoes not always hold for the symplectic leaves, as shown by Vancliff [52, Corollary 3.4].An explicit example is computed in [18, Example 3.10].

(d) The prime and primitive spectra of the algebras KP,Qn,Γ (k) introduced by Horton [25]

were analyzed by Oh in [40]. These algebras are multiparameter quantizations of O(k2n),and include quantum symplectic spaces Oq(sp k

2n), even-dimensional quantum euclideanspaces Oq(o k

2n), and quantum Heisenberg spaces, among others. Oh introduced Pois-

son algebra structures AP,Qn,Γ (k) on O(k2n), and constructed compatible homeomorphisms

P.primAP,Qn,Γ (k) → primKP,Q

n,Γ (k) and P.specAP,Qn,Γ (k) → specKP,Q

n,Γ (k), assuming the pa-

rameters involved in P , Q, Γ are suitably generic [40, Theorem 4.14].

As stated in Remark 9.2(b), a quantized coordinate ring may belong to some flat familiesfor which Conjecture 9.1 holds and also to others for which it fails. We outline Vancliff’sexample [52, Example 3.14] illustrating this phenomenon.

9.9. Example. (a) Let ai = i− 1 for i = 1, 2, 3, and set

R0 = C[h][(1 + aih)−1 | i = 1, 2, 3]

A = R0〈x1, x2, x3 | xixj = rijxjxi for i, j = 1, 2, 3〉,

where rij = (1 + aih)(1 + ajh)−1 for all i, j. This defines a flat family of C-algebras,

whose semiclassical limit is the polynomial ring R = C[x1, x2, x3] with the Poisson bracketsatisfying

{x1, x2} = −x1x2 {x1, x3} = −2x1x3 {x2, x3} = −x2x3 .

It follows from [52, Corollary 3.4] that the symplectic leaves in C3 for this Poisson structureare algebraic; hence, they coincide with the symplectic cores (Theorem 7.1(b)). By [13,

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32 K. R. GOODEARL

Example 4.6], R satisfies the Poisson Dixmier-Moeglin equivalence, and so Lemma 9.3implies that the space of symplectic leaves in C3 is homeomorphic to P.primR.

The Poisson-primitive ideals of R are listed in [52, Example 3.14] (where they arelabelled “maximal Poisson ideals”). They consist of

(♦) • the maximal ideals 〈x1−α, x2, x3〉, 〈x1, x2−β, x3〉, 〈x1, x2, x3−γ〉, for α, β, γ ∈ C;• the height 1 primes 〈x1〉, 〈x2〉, 〈x3〉, and 〈x1x3 − λx22〉, for λ ∈ C×.

We can compute them by using §9.6(b) to find the Poisson prime ideals of R and thenapplying the Poisson Dixmier-Moeglin equivalence. For instance, the Poisson center of thelocalization C[x±1

1 , x±12 , x±1

3 ] is C[(x1x−22 x3)

±1], from which it follows that any nonzeroPoisson prime ideal of R must contain either one of the xi or else x1x3 − λx22 for someλ ∈ C×. The full list of Poisson prime ideals of R is

• 〈x1 − α, x2, x3〉, 〈x1, x2 − β, x3〉, 〈x1, x2, x3 − γ〉, for α, β, γ ∈ C;• 〈x1, x2〉, 〈x1, x3〉, 〈x2, x3〉;• 〈x1〉, 〈x2〉, 〈x3〉, 〈x1x3 − λx22〉, for λ ∈ C×;• 〈0〉.

Inspection immediately shows that the locally closed points of P.specR are the Poissonprime ideals listed in (♦).

(b) A generic member of the flat family given by A is Bq = A/A(h − q)A, where q isa complex scalar such that 1 + q and 1 + 2q generate a free abelian subgroup of rank 2 inC×. For such q, the primitive ideals of Bq, as stated in [52, Example 3.14], consist of

• 〈x1 − α, x2, x3〉, 〈x1, x2 − β, x3〉, 〈x1, x2, x3 − γ〉, for α, β, γ ∈ C;• 〈x1〉, 〈x2〉, 〈x3〉, 〈0〉.

These can be computed by finding the prime ideals using §9.6(a) and then applying theDixmier-Moeglin equivalence, which holds because Bq is a quantum affine space [15, Corol-lary 2.5].

Observe that primBq is not homeomorphic to P.primR. For instance, primBq has ageneric point, while P.primR does not.

(c) In contrast to the above, any generic Bq is a member of a flat family with asemiclassical limit R′ (the algebra R, but with a different Poisson structure) such thatprimBq ≈ P.primR′ and specBq ≈ P.specR′, by [18, Theorem 3.6].

It would be very interesting to obtain criteria to determine which flat families yield“good” semiclassical limits relative to Conjecture 9.1. For quantum affine spaces andtheir Poisson analogs, one good condition appears in the work of Oh, Park, and Shin [41,Theorem 3.5] – roughly, if the scalars appearing in the defining Poisson brackets of thesemiclassical limit arise from an embedding into k+ of the subgroup of k× generated bythe scalars appearing in the defining commutation relations of the quantum affine space,then the prime and primitive spectra of the quantum affine space are homeomorphic tothe Poisson prime and Poisson-primitive spectra of the semiclassical limit.

We close with an example of the “simplest possible” quantum group for which primitiveideals match symplectic cores but not symplectic leaves. There is no nontrivial multiparam-eter version of quantum SL2, and to deal with Oq,p(SL3(C)) would require investigating36 families of primitive ideals (indexed by S3 × S3, as in [24, Corollary 4.5]). Instead, we

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 33

look at a multiparameter quantization of GL2. It is convenient to use Takeuchi’s originalpresentation [49].

9.10. Example. For the classification of primitive ideals, we assume only that k is alge-braically closed, and we choose a generic pair of parameters p, q ∈ k×, meaning that theygenerate a free abelian subgroup of rank 2 in k×. We restrict to k = C and special choicesof p and q when setting up a semiclassical limit and discussing symplectic leaves.

(a) Define the two-parameter quantum 2× 2 matrix algebra Mq−1,p as in [49]. This isthe k-algebra with generators X11, X12, X21, X22 and relations

X11X12 = qX12X11 X11X21 = p−1X21X11

X21X22 = qX22X21 X12X22 = p−1X22X12

X12X21 = (pq)−1X21X12 X11X22 −X22X11 = (q − p)X12X21 .

The element D = X11X22 − qX12X21 is the quantum determinant in Mq−1,p, but it isnormal rather than central:

XijD = (pq)i−jDXij

for all i, j [49, §2]. Since the powers of D form an Ore set, we can construct the Orelocalization A = Aq−1,p = Mq−1,p[D

−1]. There is a Hopf algebra structure on A [49, §2],but we do not need that here.

For comparison with other presentations of multiparameter quantized coordinate rings,we point out that A = Opq−1,p(GL2(k)) in the notation of [12, §1.3; 4, §I.2.4]), wherep =

[1 q−1

q 1

]. In particular, [4, Corollary II.6.10] applies, implying that all prime ideals of

A are completely prime.Observe that X12 and X21 are normal in A, and so we can localize with respect to their

powers. Although X11 is not normal, its powers also form an Ore set (e.g., verify this firstin Mq−1,p, which is an iterated skew polynomial ring over k[X11]). Note that any ideal Iof A which contains X11 also contains X12X21, whence D ∈ I and I = A. Hence, no primeideal of A contains X11, which means that no prime ideals of A are lost in passing from Ato the localization A[X−1

11 ].(b) The quotient A/〈X12, X21〉 is isomorphic to a commutative Laurent polynomial

ring k[x±111 , x

±122 ]. Hence, we know the prime ideals of A containing 〈X12, X21〉. The others

correspond to prime ideals in the localizations(A/〈X12〉

)[X−1

21 ],(A/〈X21〉

)[X−1

12 ], and

A[X−111 , X

−112 , X

−121 ]. We claim that these localizations are simple algebras, from which it

will follow that the only prime ideals of A not containing 〈X12, X21〉 are 〈X12〉, 〈X21〉, and〈0〉.

First,(A/〈X12〉

)[X−1

21 ] is isomorphic to the algebra

T1 := k〈x±1, y±1, z±1 | xy = p−1yx, xz = xz, yz = qzy〉.

Via §9.6(a), we compute that Z(T1) = k, whence T1 is simple. Similarly,(A/〈X21〉

)[X−1

12 ]is simple.

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34 K. R. GOODEARL

Third, observe that X22 = X−111 (D + qX12X21) in A[X−1

11 ], and so this algebra can begenerated by X±1

11 , X12, X21, D±1. Consequently, A[X−1

11 , X−112 , X

−121 ] is isomorphic to the

k-algebra T3 with generators x±1, y±1, z±1, w±1 and relations

xy = qyx xz = p−1zx xw = wx

yz = (pq)−1zy yw = (pq)−1wy zw = pqwz.

Another application of §9.6(a) shows that T3 is simple, establishing the claim.Therefore, the prime ideals of A consist of

• the maximal ideals 〈X11 − λ, X12, X21, X22 − µ〉, for λ, µ ∈ k×;(♦) • the ideals 〈X12, X21, f(X11, X22)〉, for irreducible polynomials f(s, t) ∈ k[s±1, t±1];• the ideals 〈X12, X21〉, 〈X12〉, 〈X21〉, and 〈0〉.

(c) The torus H = (k×)4 acts on A by k-algebra automorphisms such that

(9.10c) (α1, α2, β1, β2).Xij = αiβjXij

for all i, j. Only four of the prime ideals of A are H-stable, and thus [16, Corollary 2.7(ii),Remark 5.9(i)] implies that A satisfies the Dixmier-Moeglin equivalence (cf. [4, CorollaryII.8.5(c)]). Therefore, the primitive ideals of A are

• the maximal ideals 〈X11 − λ, X12, X21, X22 − µ〉, for λ, µ ∈ k×;• the ideals 〈X12〉, 〈X21〉, and 〈0〉.

(d) Now restrict to k = C, choose α ∈ R \ Q, assume that q is transcendental overQ(α), and take p = 1 + α(q − 1). The assumptions on α and q ensure that the subgroup〈p, q〉 ⊆ C× is free abelian of rank 2, as needed above. Our choice of p is a first-orderTaylor approximation of qα, which is convenient for extension to polynomial rings.

Choose a Laurent polynomial ring k[z±1], set zα = 1 + α(z − 1), and let B = Mz−1,zα

over k[z±1, z−1α ] in the notation of [49, §2]. Thus, B is the k[z±1, z−1

α ]-algebra given bygenerators X11, X12, X21, X22 and relations

X11X12 = zX12X11 X11X21 = z−1α X21X11

X21X22 = zX22X21 X12X22 = z−1α X22X12

X12X21 = (zzα)−1X21X12 X11X22 −X22X11 = (z − zα)X12X21 .

Observe that B is an iterated skew polynomial algebra over k[z±1, z−1α ], and so it is tor-

sionfree over k[z±1]. This algebra has been arranged so that B/(z − q)B ∼= Mq−1,p andB/(z− 1)B ∼= O(M2(k)). In B, the quantum determinant is D = X11X22− zX12X21, andit is normal. We set C = B[D−1] and observe that C is a torsionfree k[z±1]-algebra suchthat C/(z − q)C ∼= A and C/(z − 1)C ∼= R := O(GL2(k)).

Thus, A is one of the quantizations of R in the family of algebras C/(z − γ)C. Thesemiclassical limit of this family is the algebra R, equipped with the Poisson bracket

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SEMICLASSICAL LIMITS OF QUANTIZED COORDINATE RINGS 35

determined by

{X11, X12} = X11X12 {X11, X21} = −αX11X21

{X21, X22} = X21X22 {X12, X22} = −αX12X22

{X12, X21} = −(1 + α)X12X21 {X11, X22} = (1− α)X12X21 .

To find the Poisson prime ideals of R, we can proceed in parallel with part (b) above,using §9.6(b) in place of §9.6(a). We compute that the Poisson prime ideals of R canbe listed exactly as in (♦). This yields an obvious bijection φ : specA → P.specRgiven by “preservation of notation”. Clearly φ and φ−1 preserve inclusions, and so φis a homeomorphism by Lemma 9.4(a). (Alternatively, one can easily identify the closedsets in specA and P.specR and then check that φ and φ−1 are closed maps.)

The torus H acts on R by Poisson algebra automorphisms satisfying (9.10c), and onlyfour Poisson prime ideals of R are stable under this action. Consequently, [13, Theorem4.3] implies that R satisfies the Poisson Dixmier-Moeglin equivalence. Thus, the Poisson-primitive ideals of R are

(♠) • the maximal ideals 〈X11 − λ, X12, X21, X22 − µ〉, for λ, µ ∈ k×;• the ideals 〈X12〉, 〈X21〉, and 〈0〉,

and therefore φ restricts to a homeomorphism primA→ P.primR.(e) In view of (♠), we can now identify the symplectic cores in GL2(C) ≈ maxspecR

with respect to the Poisson structure under discussion. They are

• the singletons

{[λ 00 µ

]}, for λ, µ ∈ C×;

• the sets

[C× C×

0 C×

],

[C× 0C× C×

]and

{[λ βγ µ

]∈ GL2(C)

∣∣∣∣ β, γ 6= 0

}.

The space of symplectic cores in GL2(C) is homeomorphic to P.primR by Lemma 9.3.

Since

[C× C×

0 C×

]and

[C× 0C× C×

]are complex manifolds of odd dimension, they cannot

be symplectic leaves. In fact, each is the union of a one-parameter family of symplecticleaves, which can be calculated as in [18, Example 3.10(v)]. For instance, the symplectic

leaves contained in

[C× C×

0 C×

]are the surfaces

{[λ β0 δλα

] ∣∣∣∣ λ, β ∈ C×

},

for δ ∈ C×.

Acknowledgements

We thank J. Alev, K. A. Brown, I. Gordon, S. Kolb, U. Kraehmer, S. Launois, T. H.Lenagan, E. S. Letzter, M. Lorenz, M. Martino, L. Rigal, M. Yakimov, and J. J. Zhangfor many discussions on the topics of this paper.

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36 K. R. GOODEARL

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38 K. R. GOODEARL

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